A few stand out in my memory that I enjoyed. One was a World's Finest story where she and Superman team up against Batman and Supergirl. Neal Adams drew it and I think it involved an alien posing as actor. Another is her debut in Detective Comics.

As I have mentioned many times here at CCF, when I was a kid reading comics Batgirl was my favorite DC character (along with some of the Legionnaires). I just loved the the idea that an ordinary person could become a costumed crime-fighter by dint of smarts and athletic prowess.  I had her debut issue (Detective #359) but thanks to spotty distribution I missed a lot of her subsequent appearances. I did manage to get World's Finest #169. That became one of my all-time favorite comics--Curt Swan drew a great Batgirl--and I even wrote a letter (that was published) applauding the Supergirl-Batgirl team-up. They re-teamed in WF #176, this time the art was by Neal Adams, so you'll probably want to read that issue even though the story (by Cary Bates) is silly.

Then when Batgirl got her solo series in Detective, and even though she was alternating the back-up spot with Robin, I was in seventh heaven. I only read comics for a few more years, during which time I did read a lot of her solo Detective stories. The stories were kind of simplistic but the great Kane art--usually inked by Anderson--made the series special.  

As an adult I bought the Batgirl Showcase volume and it filled in the blanks, i.e., issues I'd not read back then. Hoosier X , I know you hunt down the actual old comics themselves but you might also want to pick up the Showcase volume as a guide. It contains all of her Silver Age appearances/early Bronze Age appearances up to 1974.  
dc.fandom.com/wiki/Showcase_Presents:_Batgirl_Vol._1_(Collected)

I used to have this one, Detective Comics #397, and remember the Batgirl back-up story 'The Hollow Man' had Gil Kane & Murphy Anderson art, one of my favorite combinations... the story is: good-looking guy pretends to be disfigured guy and kills women for being fickle and liking him only for his good looks, has date with Barbara Gordon!

That Batgirl?

Hmm. Oh.

I always liked Oracle better. A heroine who went through hell and then uses her resultant disability to be a better superhero feels very inspiring, and very original. John Ostrander's piece-by-piece rebuilding of Barbara Gordon in Suicide Squad is one of my favorite things in superhero comics ever.

And I tend to prefer the original versions of super-types, so Betty Kane is always my favorite Batgirl. I'm not saying the pre-new-look Batman stories were classics for the ages, but I liked the combination of mystery and sexual tension between Bat-Girl/Robin and Batwoman/Batman.

Batman 311 is one of those comics I fondly recalled having as a kid and appreciate even more as an adult.  It's one of those tightly written stories which upon reflection, seems difficult to believe could be contained within a mere 17 pages.  Early on in the tale, Englehart has Batman and Commissioner Gordon acknowledge that given their relationship, Barbara Gordon may as well be Batman's niece.  After all, how many nights did she spend watching as he and her father planned all sorts of strategies for taking down criminals in her family home?  It's a moment which could exist on its own as a nice summation of why it was inevitable that Barbara Gordon chose the path she did, but here it pays out in dividends as Batgirl is presented as a hero up against incredible odds, but at no point is she in over her head.  She's a master tactician who, unlike Batman and Robin, is completely self-taught.  Interestingly, Batman tracks down the villain of this issue, Dr. Phosphorous only to be taken out of the game early on in the issue's final battle.  Now, Phosphorous could kill you simply by touching you and Batgirl knows this.  Therefore, her best bet is to maintain a safe distance from him at all times, except the further she gets from Phosphorous, the closer Phosphorous gets to an unconscious Batman laying about ten feet away.  You can't help but realise that you're watching a master tactician at work when Batgirl's every move in this battle is both offensive and defensive.  Batman only wakes up after the battle is over in fact.

I never read any Batgirl before the Gail Simone... I like both that version and the 'Burnside' reboot. I have the showcase volume, which is pretty fun, too.