Hiya! My knowledge mostly comes from the west side of Leeds, so I'll advise on the area I know. You'll have to find someone else to advise on the east.
If your budget can afford it then Pudsey is a nice place with good links to both Leeds and Bradford, via the train and buses. It's got parks and a nice shopping centre, but I can't really advise on the schools and clubs as I don't have any children - yet! It's a little further out from Leeds city centre though - not sure if that's good or not?
If you want things a bit cheaper, then Bramley isn't too bad. It's where I'm based, and I have no major complaints. It used to be a bit rough in places, but they've been making a lot of improvements and it's become a much nicer place in the last few years. The prices have yet to catch up, which is its main bonus. It's also on the train line to both Leeds and Bradford, though the trains don't stop at the station quite as often as at Pudsey. It has a shopping centre too, with a reasonable sized Tesco and a post office.
Avoid places like Beeston and that area, they're reputed to be a bit rough and the connections aren't great. Basically, aside from Pudsey is generally best to avoid anything South of the railway line - which is the advise I was given when I moved to Leeds.
Headingley might be an option if you're studying, as it's near to one of the campuses and got shops and facilities. However, because of that it's absolutely packed with students in their late teens/early twenties, with all the associated issues that brings. Every second shop there is a fast food place or an off license, for example. Near the cricket stadium can also go manic busy on days when they're playing.
A little further out, Rodley and Calverley are really nice areas, but they aren't great for transport links. They have some buses, but no trains. The schools there are meant to be very good though.
I hope that helps a little.