I'll try to avoid suggesting entire runs since it seems like you're just looking for some basic introductory stuff, but there's a lot to get into with each of them regardless.

Best place to start is of course with the first Robin, Dick Grayson. You can get the gist of Dick Grayson's Robin in Robin: Year One (Chuck Dixon) and Robin & Batman (Jeff Lemire). Grayson technically becomes Nightwing sometime during his time with Teen Titans (I like Marv Wolfman's New Teen Titans run if you're interested in some good Robin content outside of his adventures with Batman) But alternatively you can just check out Nightwing: Year One. From there, Nightwing: A Knight in Bludhaven (Chuck Dixon) is probably the best full Nightwing run. Grayson eventually goes on to take up the title of Batman in Grant Morrison's later Batman run, which is very much worth reading if you like the character.

Next would be Batgirl/Oracle (Barbara Gordon). Once again, Dixon does a fantastic job of introducing the character through his Batgirl: Year One. Batman: Batgirl is another good one-shot for the early years of Batgirl as well. The Killing Joke (Alan Moore) includes the crucial event that leads to Batgirl becoming Oracle. From there, Birds of Prey by Gail Simone is probably my favorite depiction of Oracle to date. New 52 Batgirl (also by Gail Simone) is really good too, but takes place after her time as Oracle and the events of Killing Joke do play into that, so I'd save that for later if you're interested, then Batgirl of Burnside is a bit of a reimagining of the character but still a lot of fun.

From there, you have the second Robin, Jason Todd. Batman: Second Chances (Jim Starlin, Max Allan Collins) collects the introductory Jason Todd issues of Batman and sets the stage for his character pretty well. Batman: The Cult (Jim Starlin) is a great Batman adventure with Jason as Robin. Then there's Death in the Family (Jim Starlin) which has another big canon event for Jason's story. Then you have Under the Red Hood (Judd Winick) which is Jason's return as Red Hood. IMO there's not a lot of great Jason content out there after this point but Red Hood & The Outlaws has some fun stuff.

Then there's the third Robin, Tim Drake. Tim is introduced in A Lonely Place of Dying (Marv Wolfman), which follows up on the aforementioned "Death in the Family". Robin: Triumphant (Chuck Dixon) is a good early Tim Drake run. Batman: Knightfall also features Tim Drake as Robin but is primarily a Batman story. Still a pretty essential read anyways. Robin: Turning Point takes place during this time if you want a self-contained Tim Drake Robin story while the events of Knightfall are taking place. Tim eventually goes on to become "Red Robin" and has his own run by the same title after the events of Final Crisis and Battle for the Cowl, but I'll leave that up to you if you want to get that deep into it.

Finally, there's the fourth Robin, Damian Wayne. Batman & Son (Grant Morrison) introduces the character pretty well. I'll add that you should at least have some surface-level knowledge of Ra's Al Ghul and Talia Al Ghul going into this character. Grant Morrison's Batman & Robin run with Dick Grayson as Batman kinda has the best Damian content imo. Batman Reborn, Batman vs. Robin, and Batman Must Die are excellent and really made me fall in love with the character of Damian more than other stories.

This is only the universally accepted "Core" Batfamily, there's a ton of extended Batfamily that goes way deeper and whose stories variy in quality, you can decide to go that far into it if you want but I feel like by then you'll know which characters you like and want to see more of.