I'll denote the better/recommended stories with a *.
"The War Games"* is a strong outing for their first appearance, which is shockingly limited in terms of minutes on screen. But we see who and what they are and a glimpse of their power. Holds up well...
The next time they're used in any big way is "The Three Doctors", albeit as un-powerful as anything could get. But they do play a pivotal role, considering the Doctor's lore up to that point. "Omega" is introduced and is a cliche mustache-twirling baddie, complete with loud voice and of the sort that had everyone under the age of 5 being scared enough to wet their pants. Anyone over 3, not so much...
After that, "The Deadly Assassin" features them doddering about - also just as
poorly, if not more so. And to the point some in the audience were cheering the Master on. The same story also introduces a few cringe inducing jokes and, of course, the infamous regeneration limit. This was allegedly going to be built up for the Doctor further on except Hinchcliffe left the show, so the indirect revelation was left hanging, and used by later producers like John Nathan-Turner (and Steven Moffat) to build upon. Until this story, the Doctor does say he's regenerated a "a couple times before" (which contradicts the Morbius story, which also keeps it woolly as to whether or not the Doctor had pre-Hartnell incarnations or if it was a mind game as both Morbius and the Doctor were still mentally strong enough to fight) but I'll get back to that in a moment...)
"The Invasion of Time" is the next time we see them in any capacity, and it's set on their planet again. Except this time it doesn't matter how many companions the Doctor has, they're all allowed to trundle on in and party with him. And that's the one aspect of the story that was given any attention to detail.
Up next is "Arc of Infinity" from 1982 - again, set on their home planet and are used in any actual capacity. Unfortunately, we get the cliche 'one amongst us is a villain, moohahaha" trope and after a while it's not hard to guess. But it's got a bit of action and intrigue, and Maxil constantly steals the show. Definitely an improvement from anything made in the 1970s... Omega makes a return. There is zero continuity between this and
T3D above, which feels just wrong. That aside, Omega is treated like a proper character. Add in 1982 new wave hairdos with bright yellow socks and I honestly wouldn't skip on this despite its foibles.
Next up is "The Five Doctors"*. Yep, they're back on Gallifrey and "Rassilon" is introduced. As with "The War Games", the same author penned this one and the depth and quality of his use of his creation shows. He would say decades later he disliked how he wrote them in Classic Who, which is strange as they're so much better handled by him than most other writers using them. This story single-handedly elevates the Time Lords back to "proper mystery" status.
"The Trial of a Time Lord" is their final use on the show. Pretty much as boring as you'd expect, at least the Valeyard is an interesting creation that's also extremely well-acted. Robert Holmes had the majority of their inclusion and the results are largely good, though his first four episodes feel more derivative than they should.
And now, the tangential stories where they're mentioned or hinted at.
"Terror of the Autons"* sees a Time Lord coming to visit the Doctor in exile. He's played up a bit camp with the lame joke about his coordinates being off, but ignoring that introduction he's used to decent if not limited effect. He works well off of Pertwee. The same story introduces the Master, whose portrayal by Roger Delgado would ensure more than success from the get-go. Robert Holmes' penchant for sharp character dialogue is just as strong... What's funny is, Holmes pretty much took his previous year's "Spearhead from Space" and retooled it for the sake of the Master. Without the Master, this story is a total flop. Note how I gave it the "recommended" status, that's how great the Master is.
"The Claws of Axos" hints that the Time Lords allowed his TARDIS to operate (remote control) so he could trap the Axons, but that's about it. It's still a great story, but a one-off sentence at the end explaining their use isn't enough in terms of being a Time Lord-centric story.
"Colony in Space" has the Time Lords actively manipulating the Doctor but they're used solely as plot narrative bookends. The Master too sets up a big caper, hence the need to get the Doctor involved.
"Planet of the Spiders" introduces Cho-Je, a Time Lord who's also incognito on Earth and helps the Doctor regenerate. He was once an associate of the Doctor's.
"The Brain of Morbius" is all about Gallifreyans and alliances, and an enemy, but is remarkably tangential. (Oh, I'm sorry for that overly long moment, I myself went through two bathroom breaks and a double cheeseburger with a redbull big gulp in between.) Terrance Dicks called it bland, hence using a pseudonym. It's heavily inspired by "Frankenstein" but adds so much more into the lore as well as altered premises that anyone unaware would not think much about it. It's a rather clever setup.
"State of Decay"* involves a long-past war with "The Great One" - a particularly nasty vampire, and one that almost took over the entire galaxy. Terrance Dicks wrote it three years earlier but the BBC ordered it put aside because they didn't want it to air around the time their non-scifi adaptation of "Dracula" was being made. I loved this one as a kid and a lot of it holds up.
"Resurrection of the Daleks" has the Daleks scheming to do about eight million things for flimsy reasons, but dang if it's not action-packed... But one of those eight million things is to invade Gallifrey to assassinate the High Council, using the Doctor to set the stage as he accepted the Lord President position on Gallifrey after "The Five Doctors", though there's no direct or indirect confirmation of that piece of continuity. (Fans who sat through Davison's era probably will connect the dots, but not necessarily, and it's not a necessity that picking up on that plot detail was intended - or even needed.) It does elevate the Daleks a little in terms of being an actual foe, and not as wind-up tin robots, since their previous outing used them as a joke...
"The Twin Dilemma" sees another former associate and friend, Azmael returning. Some say he was to be Cho-Je, but the regeneration count stated on screen is technically off by 1. Maurice Denham really puts in a robust and believable performance for the plight of Jaconda and himself. The story was written by an accomplished writer, Anthony Steven, who had little sci-fi background so Eric Saward had to take over - and his two episodes fare so much better considering the problems the story had (and the fact there was no time to start over). The story is such a garbled mess, but has some interesting sci-fi ideas at work and Azmael's plight is worthy enough to keep watching. That, and Colin Baker works overtime to make some stinky dialogue (and direct continuity from the previous story) work.
"Attack of the Cybermen" hints at the Time Lords maneuvering the TARDIS to get the Doctor to meddle with the Cyber scheme to change history. Part 2 has its moments, but part 1 is a lot more solid. The Doctor throws a fit when realizing he was maneuvered. Of all the incarnations, the 6th had the most interaction with the Time Lords behind the scenes and in some ways it's rather good.
"The Two Doctors" had author Robert Holmes mistaking the third as the second Doctor, which had fans creating 'Season 6B" to canonize the resultant problems after getting bored of griping of the continuity problems. Again, Time Lord interference is hinted at and this time it's about illegal time travel experiments setting the stage. Robert Holmes, who wrote this to take place in Louisiana but then had to rewrite it hastily because the US partner in financing the story pulled out and anything is up for fun speculation. That said, the end result due to the rewrite (now taking place in Seville, Spain - yum!) loses some of the original banter between the Doctor and Peri, and to be honest the Sontarans are not needed as there's enough plot for everyone else, but he also said he wanted to improve their status after "The Invasion of Time"... I like the story but most keep it in the "mediocre" range. It could have been more and I wish the original scripts were available...
The original, unmade season 23 had little in development... but its finale, "Gallifrey", would have featured the Doctor's home planet (again). Speculation suggests the planet might have been destroyed, but that's neither here nor there: Considering the 6th Doctor's era was peppered with a lot of dialogue regarding the Time Lords and their alleged involvement in events, I wish something proper had been developed before the show was put on hiatus and retooled but the fact enough information existed to say this would have been the finale only whets the appetite for fans of Colin's era (like meeeeeee!

). They clearly were working to a plan for Colin's era with all this. Okay, that's speculation but the theory fits. Or doesn't not fit.
Side note to the previous paragraph: A lot of the unmade season 23 is definitely better than the retooled and produced season from 1986 (though the new theme music has a great haunting tone, even if synth sax instruments hadn't been perfected yet - the idea behind the composition is rather great, but I digress.) And most of those were sufficiently completed to actually be released in some form. Three were novelized as little work had to be done. That said, most of the season had enough in terms of approved synopses that they were cobbled into audio releases - these were largely good but there's something about the novels that did them more justice... "Gallifrey" was not amongst these, nor was the story with the presumed/unconfirmed title of 'Yellow Fever and how to Cure it" - a story set in Singapore for some reason and (speculatively) featured the Master and Autons and possibly the Rani, who first appeared in "The Mark of the Rani" and Kate O'Mara simply stole the show as the newest rogue Time Lord, that of amoral scientist using the galaxy as a giant petri dish. Very original and memorable character, thanks in no small part to Kate. (Her second appearance in the show didn't fare as well, but that's another story...)
"Remembrance of the Daleks" has the Emperor spouting on and on about "its impotent column of Time Lords" in the most bizarre double entendre of all time, and the rest of that scene was truncated since the bit where the Doctor boasts about how he's more than just another Time Lord was deemed "too excessive and insincere" and was excised from being filmed, but it's an extra on the DVD. The same (save for no deleted bits as DVD extras) also gets to be said for...
..."Silver Nemesis", whose original intent was to reveal the Doctor as being a god - but the producer, who still allowed the story to be made, demanded this be toned down significantly because they were trying to introduce mystery, not demystify with a sappy, lame explanation. And rightly so. He let script editor Andrew Cartmel have some creative freedom but still had final say. Given that everyone felt he would curtain scenes from "The Curse of Fenric" but didn't, the fact he did curtail this attempted revelation is that much more significant as a result. The story involves another Time Lord artifact as well as a plot similar to "Remembrance of the Daleks" (not unlike how "Resurrection of the Daleks"'s plot is similar to "Earthshock"'s.) The Doctor starts spouting stuff about prototypes and namedrops Rassilon and Omega as if they were all a bunch of happy drunken barflies on a Friday night. The scene between Peinforte and Ace at the end about this faux mystery of the Doctor is nicely done, but Fiona Walker (Lady Peinforte) pretty much steals the show in her own right. The story is limited to 3 episodes when 5 were needed given everything it's juggling, but roll with its issues and it's still reasonably good.