I'm referring to Picard, Riker, Data, Worf, Geordi, Dr. Crusher, Wesley Crusher, Chief O'Brien and family, Deanna Troi, Guinan. I know Picard and Worf show up on DS9, though I haven't seen it myself. I'm curious what happened to the characters after that last episode of TNG?
Miles O'Brien: Ends up transferring to DS9 after accepting a posting as Chief Operations Officer for the station. His wife, Keiko, and daughter, Molly, follow him and live on DS9 as well. Appears in multiple seasons of DS9.
Lt. Cmdr Worf: Transferred to DS9 to Command Division following destruction of the Enterprise-D (Star Trek: Generations film). Eventually accepted position as Federation ambassador to Qo'nos. Appears in multiple seasons of DS9, as well as multiple TNG-era films.
Wesley Crusher: after resigning his commission with Starfleet, he opts to live on Dorvan V to learn from "The Traveler" to journey to other planes of existence. Appears very briefly in Star Trek: Nemesis in a Starfleet uniform, at the wedding. (A cut scene includes dialog with Picard.)
Dr. Beverly Crusher: Serves as Chief Medical Officer on Enterprise-E under the command of Cpt. Jean Luc Picard. Does not appear on any other Star Trek series, but does appear in the TNG-era films.
Cpt. Jean Luc Picard: Appears on DS9 a few times in the beginning season. Still in command of Enterprise-E after the loss of the D vessel (Star Trek: Generations), and appears in all TNG-era films. Name is mentioned during Voyager.
Lt. Cmdr. Data: Did not appear on DS9, although did interact with Dr. Julian Bashir from DS9 while Enterprise-D was in orbit of the station during a TNG episode. Appears in all TNG-era films.
Killed while in service aboard Enterprise-E, by sending Picard back from the exploding Reman vessel instead of himself, sacrificing his life and ending his career.
Lt. Cmdr. Geordi La Forge: Serves aboard Enterprise-E as Chief Engineer. Does appear in Star Trek: Voyager in command of his own vessel in an alternate future timeline. Also appears in the TNG-era films.
Cmdr. William Riker: Continued to serve as First Officer to Cpt. Jean Luc Picard even during DS9 until after the events of Star Trek: Nemesis. Eventually leaves the Enterprise-E to command the USS Titan as Captain. Appears in the TNG-era films, in Enterprise (holodeck episodes) and once on Voyager (thanks to Q).
Guinan: Serves as bartender in Ten-Forward on Enterprise-D. Does not appear in any other Star Trek series, except the TNG-era film Generations. Not clear what Guinan does after the D is destroyed.
Cmdr. Deanna Troi: Continues service aboard Enterprise-E as Ship's Counselor. Appears in the TNG-era films. Appears in Voyager and in Enterprise (holodeck episodes). Eventually marries Cpt. William Riker and follows him to the Titan where they have a child.
Q: Included due to his multiple appearances across multiple series. Q appears quite frequently in TNG, Voyager, and DS9. Spreading his mischief and exploring what it means to be a "petulant" human from a Q's perspective, Q simply IS. Even after TNG, he IS. Period.
Thomas Riker: Created by a freak transporter accident years before Will Riker's posting on the Enterprise-D, Thomas Riker eventually resigns his Starfleet commission and enters into rebellious alliance with the Maquis. He appears in DS9 during the episode "Defiant", where he surrenders his Maquis ship and crew, spending the rest of his life (presumably) in a Cardassian labor camp.
Lt. Reginald Barclay: Serves aboard USS Enterprise-D and E. Eventually transferred back to Earth to San Francisco to work on the Pathfinder Project.
Ends up contributing majorly to opening communications to Voyager in the Delta Quadrant and helping them get back to Earth.
Here is everything I can remember, written to avoid spoilers as much as possible.
Captain Picard was in "Emissary", the pilot episode of DS9.
The Enterprise stopped at DS9 in "Birthright Part 1" and so several DS9 characters were guests in that TNG episode, which I guess is the reverse of what you asked about.
Worf was a DS9 regular character in the 4th, 5th, 6th, and 7th seasons. Miles O'Brien and his family moved to DS9 in "Emissary", and O'Brien became noted for the many bad things which happened to him in various episodes.
A duplicate of Will Riker appeared in a DS9 episode,
One of the regular TNG characters was killed during the first season but sort of kind of returned in a few later episodes, one of the regular TNG characters was replaced by another one during the second season and then returned in the following seasons. Wesley Crusher went to Starfleet Academy but returned in a few episodes and made a major life decision in one of them. At Starfleet Academy Wesley and some fellow cadets were involved in a scandal. One of those cadets later served on the Enterprise in one episode, while many fans think that another one changed their name and was a regular on Voyager.
"All Good Things" the last episode of TNG, saw Q return for a final confrontation with Picard, and Picard found himself jumping in time from the time of "Encounter at Farpoint" to his present and to 25 years in the future of "All Goods Things", and saved all life on Earth or more.
So "All Good Things" showed what the future of the characters would be for 25 years in the future. Except that it showed one possible future out of many (see the episode "parallels" to see a multiverse of many alternate universes), and two TNG movies contradicted it in a couple of major details, thus showing that the TNG movies and the 25 years in the future scenes are two separate and different possible futures of "All Good Things".
The TNG cast appeared in four movies: *Star Trek: Generations", Star Trek: First Contact, Star Trek: Insurrection, and Star Trek: Nemesis and there was a marriage and the death of a major character in one of them.
Q brought Riker to Voyager in one Voyager episode, and Deanna Troi and Reginald Barclay (an engineer in several TNG episodes) communicated with Voyager in several episodes.
And Riker and Deanna Troi appeared in "These Are the Voyages", the last episode of Enterprise, as others have mentioned.
The new Star Trek: Picard series features some of the former TNG crew, besides of course the titular character Jean-Luc Picard.
Data appears in the fisrt episode, playing Poker with Picard in a dream sequence.
He also make another and final appearance in the last episode, when he asks Picard in a "quantum simulation" dream-like sequence to have his consciousness be terminated, thus letting him be free to finally "die".
Riker & Troi
In the seventh Episode, Picard meets William Riker and Deanna Troi in their house on planet Nepenthe, where they now live with their daughter.
The last episode of Enterprise featured Riker interacting with the characters of Enterprise via a holodrama based on them. However that was supposed to be the Riker from the TNG episode "The Pegasus". What is soon supposed to happen is a sequel series streaming on CBS Online which will reveal what happens to Captain Picard after his time as captain of the Enterprise is over.