I think most people will tell you to read Chris Claremont's run on Uncanny X-Men, the whole thing. The dialogue is pretty clunky and hokey since it's quite old now, so it might be a bit weird to read but it's where the X-Men pretty much hit their stride in popularity and quality. Current stories still refer to what happened in his run so you pretty much need to know it.
If you want something a bit newer you can read Grant Morrison's run which was New X-Men 2001. Critically acclaimed series that one.
Don't really know if either are in omnibus form. Probably are though since they're so popular.
These two lists are probably helpful for X-Men. Though there is quite a lot it might help you narrow down choices a bit.
http://25.media.tumblr.com/tumblr_m8e8kgPnb21rpok63o1_1280.jpg
http://25.media.tumblr.com/tumblr_m8e8lg442f1rpok63o1_1280.jpg
from the
http://comraderecs.tumblr.com/ , a very useful place to get comic recommendations.
As to why the Avengers is so prominent in X-Books now, it's because last year they had a big Avengers vs X-Men event, so now the groups interact a lot more.