The best timeline I have seen which includes dates and times for the Fellowship of the Ring can be found at The Lord of the Rings Fanatics network ^[1]. There are four parts to their timeline and each lists the paths of the major participants in the fellowship and events related to the characters. The fine graphic created at xkcd ^[2], is more of a story-path tracker than an actual timeline.

The Lord of the Rings Timeline was designed by: Philip Kooijman © 2001.

The short answer is roughly 20 years. The passing of the ring from Bilbo to Frodo takes place far in advance of most of the story. Most of the action happens over the course of 3 years.

The long answer is distilled from this ^[1] extremely thorough chronology, here's a rundown. All events take place during the Third Age. Let me know what events you'd like me to add.

• 3001 - Bilbo Baggins turns 111. He announces his plan to leave The Shire and does so. Left behind, on purpose, is The One Ring. Frodo and Merry find the ring amongst Bilbo's possessions. Sometime late this year Gandalf suspects that Bilbo's ring is The One Ring.
• 3004-3008 - Gandalf visits Frodo periodically, presumably to keep his eye on the new Ringbearer.
• 3017 - Gollum is captured by Sauron's forces. He's tortured for information on The Ring.
• 3018
  • Aragorn captures Gollum and turns him over to King Thranduil of Mirkwood. He's held there, and questioned by Gandalf, who then leaves for the Shire.
  • Gandalf visits Frodo, and determines the ring is The One Ring. This begins the quest, Frodo and Sam make plans to depart. A few months later, they depart with Pippin joining them.
  • Orcs attack Mirkwood, Gollum escapes.
  • Merry joins the group.
  • The hobbits arrive in Bree at the Prancing Pony. They meet up with Aragorn.
  • Aragorn and the hobbits arrive in Rivendell. At the end of the year, the Fellowship of the Ring is formed and departs Rivendell.
• 3019
  • They fail to cross the mountains via the pass of Caradhras, so they travel through Moria. Gollum starts following the Fellowship. In Moria, they are attacked by Orcs and a Balrog. Gandalf is separated from the rest of the Fellowship while battling a Balrog.
  • The Fellowship goes to Lothlórien. There they meet Galadriel, and receive gifts. They depart down the great river Anduin.
  • The Breaking of the Fellowship:
    • Boromir attempts to steal the ring.
    • Sam and Frodo head to Mordor.
    • The group is attacked by Uruk-hai.
    • Boromir dies in battle defending Merry and Pippin. They are abducted, and taken towards Isengard.
    • Aragorn, Legolas, and Gimli put Boromir into a funeral boat and follow the Uruk-hai who have Merry and Pippin.
  • The Uruk-hai are attacked by the Rohirrim. Merry and Pippin escape and meet Treebeard and the Ents.
  • Aragorn, Legolas, and Gimli meet Gandalf, who is now 30% whiter. They head to Edoras. Then they head to Helm's Deep, and the Battle of Helm's Deep happens. Gandalf splits off prior to Helm's Deep and heads to Isengard.
  • The Ents attack Isengard.
  • Gandalf arrives at Isengard, and meets Merry, Pippin, and the Ents. Gandalf heads to Helm's Deep with backup of men and Huorns (ancient and powerful tree-like beings). They arrive at Helm's Deep and contribute to the victory. Gandalf, Aragorn, Legolas, Gimli, King Theoden and his company go to Isengard to meet Saruman. Saruman is imprisoned.
  • Gollum joins Frodo and Sam. They head to the Black Gate, then instead head to Minas Morgul.
  • Pippin uses the palantír, and is revealed to Sauron. Sauron mistakes him for the Ringbearer. Pippin and Gandalf head to Edoras, then onwards to Gondor. Merry swears fealty to Theoden.
  • Theoden's men gather at Dunharrow. Aragorn, Legolas, and Gimli take the Paths of the Dead.
  • Sam and Frodo are captured by Faramir's men. They are brought to Faramir, but he releases them, and they head to the path near Minas Morgul. They sneak in to Mordor, and make their way towards Mount Doom.
  • Aragorn, Legolas, and Gimli emerge from the Paths of the Dead, and start working their way towards Gondor, while defeating some of the backup for Sauron's forces.
  • In Minas Tirith, Pippin swears fealty to Denethor.
  • The Battle of the Pelennor Fields
    • Sauron's forces attack Minas Tirith.
    • Gandalf and the forces in Minas Tirith defend the siege.
    • Pippin and Gandalf save Faramir from Denethor's madness.
    • The Rohirrim, including Merry, attack Sauron's forces.
    • Aragorn, Legolas, Gimli, and the dead join the battle.
  • Aragorn heals a ton of people. Then the plan is formed to keep Sauron distracted from Frodo and Sam's progress. He, Gandalf, Legolas, Gimli, Pippin, and a host of others head to the Black Gate and meet the Mouth of Sauron and reject his terms. They then battle Sauron's forces.
  • Frodo and Gollum destroy The One Ring in Mount Doom. Frodo and Sam are rescued by Gandalf and the Great Eagles.
  • Everyone returns to Minas Tirith.
  • Aragorn is crowned King. There was much rejoicing.
  • Frodo, Sam, Merry, and Pippin head home. Saruman gets to the Shire first, and sets up shop in Bag End. He enslaves the hobbits, and waits for Frodo, Sam, Merry, and Pippin to return. They arrive and battle with Saruman, who is in the end betrayed by Wormtongue. Wormtongue in turn is killed by the hobbit forces.
• 3021 - Frodo and Bilbo depart for the Grey Havens. Gandalf and many elves go with them. Sam, Merry, and Pippin see them off.

See also The Lord of the Rings Wikia ^[2] for more information on the names and places herein.

The existing answers look good, but if you have a copy of the Lord of the Rings on you, just look in Appendix B, 'The Tale of Years'. There you will find a complete timeline of the second and third age, with the years 3018 and 3019 (where the story in LotR happened) in extra detail.

Tolkien himself made a very detailed fourteen page Synoptic Time-Scheme

This is kept at Marquette University, at shelfmark Mss-4/2/18/1a-8a ^[1]. A complete transcript of the time scheme has been published in full in Tolkien Studies 19, 2022 Supplement ^[2] as "The Chronology of the Lord of the Rings", and prior to that various transcribed excerpts have been included in The Lord of the Rings: A Reader's Companion. Additionally manuscript scans of the full time scheme were reproduced in JRR Tolkien The Art of the Manuscript and prior to that a few pages were reproduced elsewhere.

William M. Fliss, archivist at Marquette university, describes the time-scheme as follows:

As Tolkien wrote The Lord of the Rings he became preoccupied with time and distance, resulting in some archival gems. Among the manuscripts are synoptic time-schemes where Tolkien tracked the movements of different characters in the story. Resembling a spreadsheet in an age before Microsoft Excel, these schemes contain columns representing characters or groups of characters and rows representing days. Tolkien strove to synchronize the movements of characters on every day for much of the story. Tolkien seems to have created the final version of the time-scheme in the late 1940s as he completed the manuscript. This time-scheme is fifteen pages in length, but only nine of the pages have the spreadsheet format. The first two pages comprise a single listing of important dates, from the birthday party at Bag End on September 22nd until January 15th. Before I saw this time-scheme I never fully appreciated the importance of January 15th to the course of events. On that day Gandalf the Grey fought the Balrog on the Bridge of Khazad-dȗm and fell into the shadow of Moria. It was also the day when, to use an anachronism, the clock started ticking on the Fellowship. We learn from the time-scheme that when night fell on January 15th messages were sent forth from Moria to Isengard and to Mordor, setting evil forces in motion. Suddenly questions confronted Tolkien such as how long would it take for the message to reach Barad-dȗr? How exactly would it get there? When would Grishnákh and company be sent from Mordor? How quickly would they travel? The need for such a day-by-day reckoning intensified on February 26th with the Breaking of the Fellowship. Suddenly this tight group of characters that had moved together through Book II splinters in different directions, and Tolkien must track their every move.
_{Fliss, William M. (2017) "“Things That Were, and Things That Are, and Things That Yet May Be”: The J.R.R. Tolkien Manuscript Collection at Marquette University," Mythlore: A Journal of J.R.R. Tolkien, C.S. Lewis, Charles Williams, and Mythopoeic Literature : Vol. 36 : No. 1 , Article 3. ^[3]}

What follows are images of the manuscript itself, from a variety of sources. For a full readable transcript please consult The Chronology of the Lord of the Rings, pages 34-86.

• Page 1 - Sept 22-Oct 15: (Can be seen in this reddit picture from the NY Strong Museum of Play. ^[4] Also published in The Art of the Manuscript pg 133):

• Page 2 - Oct 16-Jan 15: Published in Voyage en Terre du Milieu pg 271 and The Art of the Manuscript pg 133.

• Page 3 - Jan 15-Jan 22: Published in Voyage en Terre du Milieu pg 272 and The Art of the Manuscript pg 135.

• Page 4: Jan 23-Feb 16: Published in The Art of the Manuscript pg 135.

• Page 5: Feb 17-Feb 25: Published in The Art of the Manuscript pg 136

• Page 6 - Feb 26-Feb 29: Published in Maker of Middle-earth pg 352-3, Voyage en Terre du Milieu pg 273, and The Art of the Manuscript pg 136

• Page 7: Feb 30-Mar 3: Published in The Art of the Manuscript pg 137

• Page 8 - Mar 4-Mar 7: Published in Voyage en Terre du Milieu pg 273 and The Art of the Manuscript pg 137

  

• Page 9 - Mar 8-Mar 12: Published in The Lord of the Rings: A Reader's Companion, frontispiece, and The Art of the Manuscript pg 138

  

• Page 10 - Mar 13-17: Published in The Invented Worlds of J.R.R. Tolkien: Drawings and Original Manuscripts from the Marquette University Collection pg 37, and The Art of the Manuscript pg 138

  

• Page 11: Mar 18-25: Published in The Art of the Manuscript pg 139.

  

• Page 12: Apr 8-Aug 10: Published in The Art of the Manuscript pg 140.

  

• Page 13: Aug 14-Sep 22: Published in The Art of the Manuscript pg 140.

  

• Pages 14: Oct 5-Oct 6: Published in The Art of the Manuscript pg 141.

  

Just for completeness' sake, I want to add to the already excellent answers with the LotRProject, a set of interactive charts and infographics based on The Lord of the Rings and other works in Tolkien's legendarium. They have a fantastic interactive timeline ^[1] of events synchronized with a map of Middle-earth that lets you track events by time, by protagonist or by location:

I found this nice and fairly thorough timeline ^[1]. I can't vouch entirely for its accuracy, but it seems to tally with my recollection, so it's probably good enough.

Some important milestones:

September 22, 3001 - Bilbo leaves the Shire for Rivendell.
April 13, 3018 - Gandalf tells Frodo about the ring.
Sept 26, 3018 - The Hobbits leave the Shire.
October 20, 3018 - The Hobbits and Aragorn arrive at Rivendell.
December 25, 3018 - The Hobbits leave Rivendell.
March 25, 3019 - The ring is destroyed.

So, a bit under a year after they knew Frodo had to leave, six months from them actually leaving. But nearly 18 years between Frodo inheriting the ring and its destruction.

To supplement the existing answers, not supplant them, I offer this link ^[1] to an interactive infographic of Time and Distance Traveled in The Hobbit and The Lord of the Rings. I can't share an image of the graphic here, since it is interactive, but it is pretty interesting and allows you to zoom in on individual days. The calendar used is apparently the Shire calendar, and each month has 30 days. It is a bit odd to see "February 30".

20 years. Bilbo was 131 at the end when he and Frodo left Middle-earth, it was his 111th birthday at the beginning of The Fellowship of the Ring.

According to Barbara Strachey's Journeys of Frodo ^[1] the entire round trip started and ended almost a year and a month to the day from the 23rd Sept to 26th Oct the following year — though it would appear they had 2–3 months respite at Minas Tirith after the Mt Doom moment on 25th March. I presume a year later Frodo & Co depart for Grey Havens on 21st Sept, meet Gandalf the next day and arrive at the Gray Havens 7 days later, 29th Sept.

At the end of the Return of the King film, when Frodo, Samwise, Meriadoc and Pippin return to the Shire, its said that "13 months to the day"

So... 13 months. In the film verse anyway