I've been trying to find out what series to watch if I want to get the whole Bat family experience like the whole story from Robin/Nightwing's side, the whole thing about Jason Todd, Tim Drake, Batgirl and later Damian Wayne.
First, realize there have been many different Batman animated series over the years, some going back to the 1960s and 1970s. I'll focus on the ones from the 1990s on, as they're the ones with the greatest sense of continuity/history (internally, at least).
Even then, there have been 4 distinct series (one of which has had at least 3 names); and, each of the four has its own continuity, distinct from the others. Most use some of the same characters, whose origins are usually at least based on some version of the origins in the comic books - but the underlying source material is sometimes the only common denominator.
Batman: The Animated Series (including The Adventures of Batman and Robin; both from the mid 1990s) covers the first appearances of Batgirl (not particularly similar to the comics' original version of her origin). The New Batman Adventures, (a continuation of the original series produced in the late 1990s, around the time Superman: The Animated Series was in production) gives the series' version of Disk Grayson's transition to Nightwing, introduces Tim Drake as Robin, and has much more frequent appearances by Batgirl.
Note: It's safe to assume that this series shares a continuity (for the most part) with the televised series Superman: the Animated Series, Justice League, Justice League Unlimited, and (to a lesser extent) Batman Beyond and its spinoff, The Zeta Project.
Robin (Dick Grayson) and Batgirl both appear in The Batman (2004-2008); due in part to Robin's role in the Teen Titans show, their stories and relationship to Batman is notably different than in the comics. Batgirl shows up starting in season 3, Robin in season 4.
Dick Grayson appears in a couple of episodes of Batman: The Brave and the Bold, but is not even one of the more common guest-stars.
None of these characters appear in Beware the Batman (the most recent TV series, from 2013). Barbara Gordon appears, but does not take on the Batgirl identity.
None of the actual televised animated series cover Jason Todd or Damian Wayne in a significant way.
However, in addition to the various televised series, DC has been producing direct-to-DVD animated movies as well, for quite a while now. some of these have continuity that lines up with some of the other movies; however, those are the exception rather than the rule, I believe.
Jason Todd's story is covered in the (direct-to-DVD) animated movie Batman: Under the Red Hood; this is an adaptation of this storyline from the comics.
Damian Wayne appears in the (direct-to-DVD) animated movies Son of Batman, Batman vs. Robin, and Batman: Bad Blood (which share the same continuity).
Within the DC Animated Universe, you need to watch five series and a movie to get the full story on the Bat family.
Aside from covering Batman, Batman The Animated Series covers Dick Grayson's time as Robin, as well as a smattering of Batgirl.
The next link in the chain is The New Batman Adventures. This covers more Batgirl, Dick Grayson post-Robin, and Tim Drake. You will note here that Jason Todd isn't included -- because he doesn't exist in this universe.
After this, The Justice League is the go-to series for hints at what's happening with the Bat family. It covers the general happenings leading up to the creation of a larger Justice League organization, in which the Bat family occasionally shows up.
For the purpose of setting up the bomb, Batman Beyond is next. It covers the early career of Terry McGinnis (He's not on your list by name. Spoilers.), the late career of Bruce Wayne, and covers Barbara Gordon's post-Batgirl career. It touches on Dick Grayson and Tim Drake's futures, as well.
Before starting up Justice League: Unlimited, watch the Batman Beyond movie: Return of the Joker. This wraps up Tim Drake's story.
Finally, tune in to Justice League: Unlimited, which continues the story of the larger Justice League organization and caps off with the completion of the stories of Bruce Wayne and Terry McGinnis.
There is no single animated/film/tv series that uses all of those characters.
The DC Animated Universe (DCAU) pretty well covers Robin and Batgirl through Batman: The Animated Series. Later, The New Batman Adventures covers Tim Drake, and Nightwing. Damian Wayne and Jason Todd were never part of the DCAU.
Son of Batman tells the basic story of Damian Wayne, it is continued by Batman vs. Robin and Batman: Bad Blood.
Batman: Under the Red Hood features Jason Todd, with flashbacks to him as Robin, and his subsequent death at the hands of Joker.