I basically looked at a few articles and videos that talked about the best episodes and used that to construct my own program based on what seemed interesting. I think I started with
this one from Nerdist, though I skipped the theatrical film, which most other sources seemed to agree was not worth it.
I know I used this one as well, though with more mixed results. And here's a
longer recommendation list from Kotaku in chronological order (the actual timeline of the series jumps around a lot).
The storylines I liked the most, all of which are mentioned in that first Nerdist article, were the Mortis trilogy (S3E15-17), the Umbara Arc (S4E07-10), the final story arc concerning Anakin's apprentice, Ahsoka Tano (S5E17-20), a couple of clones
almost uncovering their true purpose (S6E01-04), and Yoda learning weird things in weird places (S6E11-13).
The Mortis trilogy is probably the craziest thing in the series, and while I personally love how strange it gets, I would not blame anyone who thought it was the stupidest Star Wars content ever, so it's probably not what I would suggest watching first. Ahsoka's final plotline (until she appears again in the new season and on
Rebels) is probably the series' emotional high point, and also a really good illustration of how the Jedi Order in this era is actually kind of terrible. (Incorporating the events of this story into Anakin's ultimate betrayal in
Revenge of the Sith would have helped his character arc immensely.) The Yoda episodes also do some heavy lifting for the saga, in that they explain why there aren't hundreds of thousands of Jedi force ghosts popping up all over the place all the time, and also attempt to bridge the somewhat different personalities and philosophical views expressed by Yoda in the prequels vs. the original trilogy. They also bring back Liam Neeson and feature a cameo by Mark Hamill (not as Luke). And speaking of cameos, the Umbara Arc features a single episode directed by Walter Murch for some reason (his only directing credit aside from
Return to Oz).
I also watched most of the Darth Maul stories, which seem to be fan favorites, but they didn't do much for me. That said, he appeared a lot on the show, and his episodes form a huge chunk of what people tend to say is the best of the series, so YMMV. I think the Kotaku list I linked to above includes all of them (and pointedly
excludes the Mortis arc!). He's also all but certain to return in the new season based on what's already public knowledge of the storylines they had planned. Another fan favorite arc that I couldn't get into (and never finished) is the one where Kenobi goes undercover as a bounty hunter — possibly owing to my ongoing ambivalence regarding every bounty hunter character that has ever existed in this franchise.