Judy basic films, Tier 1: The Wizard of Oz (of course), Girl Crazy (to get one of the put-on-a-show with Mickey in there), Meet Me in St. Louis (for the Vincente/Liza connection), A Star is Born, I Could Go On Singing (her last film, and very much the persona of the latter-day Garland)
For history's sake and to get the broadest overview in the least amount of time, I'd recommend that you then proceed to: Broadway Melody of 1938 (to get Dear Mr. Gable), Love Finds Andy Hardy (to get a Betsy Booth/series film), The Clock (her first nonmusical and her only one at MGM), Easter Parade (to see her with Astaire), Summer Stock (to see her best with Gene), and Nuremberg (to get her final nomination).
There's plenty of worth, though they aren't absolutely essential to understanding her talent or impact, in these titles: Babes in Arms, Babes on Broadway, and Strike Up the Band (the other 3 put-on-a-show, non-Andy Hardy films with Mickey), Ziegfeld Girl (star ensemble film, but Judy is the one with the talent and it's a backstager so she wins), For Me and My Gal (her first with Gene), Presenting Lily Mars (she deadlifts this so-so script over her head in the name of keeping us entertained), The Harvey Girls (MGM musicals go western, and there's Angela Lansbury), The Pirate (with Gene again, in artsy-fartsy mode), In the Good Old Summertime (Pasternakian fun, and Van Johnson), A Child is Waiting (Cassavetes drama, she can hold her own with Burt Lancaster).
You can watch her segments alone in her guest star films without seeing the whole thing: Thousands Cheer, Till the Clouds Roll By, Words and Music, Ziegfeld Follies. For all of these you could fast forward anytime people start to talk. Just watch the musical numbers, they're like a primer of everything from the MGM musical's Technicolor era. You could also put her Nuremberg perf here in the "guest star" category too, though the context of seeing the whole film makes her work more powerful.
These, you get to 'em when you get to 'em: Pigskin Parade (her first feature, it's primitive and fun if you give over to it), Thoroughbreds Don't Cry, Everybody Sing, Listen Darling (has "Zing!..." a major song for her), Andy Hardy Meets Debutante, Little Nellie Kelly (with Singin' in the Rain), Life Begins for Andy Hardy, and the two animated Gay Purr-ee, and Pepe (where again she's a guest star).
Contrary to some of what was written above, the tv series from 63-64 is after the variety program format for the television medium was well honed. CBS just didn't know how to make a variety show around HER highly specific, but also seemingly all-encompassing, talent. You can see the different producer regimes (Schlatter, Jewison, Colleran variety format, Colleran concert format), and their different approaches to what the series should be. Her behavior was only part of the backstage drama, there was plenty else going on with the network and the other creatives. Watch them in the order they were taped, NOT broadcast order, it will make more sense. Check out the book that's just about the series, Rainbow's End, which also covers her tv career before that pretty well--several spectaculars that were massive tv events getting huge audiences. The Judy Frank and Dean special is exceptional television, and the Judy-Robert Goulet-Phil Silvers is fun.
She worked 45 of her 47 years and she made a lot of feature films. There's a good size TV career, plenty of radio appearances (where she sings songs she didn't do anywhere else) including non-musical roles in radio dramas, an album/recording career (mainly Decca and Capitol), and a long concert career well-documented on legit and bootleg recordings. She had a sad ending what with dying broke and alone on the toilet and everything, but she also knew the pleasure of being one of the most beloved stars in the world for decades. The quality of her body of work can't be denied. Her artistry endures. A lot of joy awaits you, OP. Let us know what you like!