Just so you know where I'm coming from with my recommendations- I read mostly Marvel, and I have just about every X-Men comic ever.
Magneto Testament (2008)- A 5 part story about Magneto's childhood in a Nazi prison camp. (The story could almost have been any kid in a Nazi prison camp and it still would have been a great read.)
Angel Revelations (2008)- A nice retelling of Angel's mutant powers revealing themselves. 5 comics.
Astonishing X-Men (2004)- Joss Whedon wrote 25 comics of X-Men and it's one of my favorite X-Men runs. Issues 1-24 and then Astonishing X-Men Giant Size #1
Woverine: Weapon X- the story of how Wolverine got his adamantium. Top notch story telling and art.
Old Man Logan- A possible future story about Wolverine. What could possibly make Wolverine a pacifist? Very well done. 8 issues total
Wolverine Origins- The whole series is good (50 issues plus a few tie-ins). Wolverine remembers his past and confronts old enemies and comes to peace with past (while kicking butt).
X-Men (1991)
#1-3- Chris Claremont wrote Uncanny X-Men for 17 years. It's hard to choose which comics to recommend, since his entire run was very good. However, this was the last story arch he'd write for X-Men for quite a while. Also, the X-Men fight Magneto, and that's always fun.
X-Men #25- Magneto pulls the adamantium off Wolverine's bones. Sick! (Unimportant side-note: first comic I ever bought.)
Any X-Men Crossover: generally lots of mutants and lots of battles- some that stick out in my mind: Mutant Massacre, X-Tinction Agenda, Phalanx Covenant, Age of Apocalypse (alternative reality), House of M/Decimation, Messiah Complex
I'm sure I could recommend more X-Men stuff, since I love just about all of it, but here's some other Marvel picks:
Captain America #357-364: "
The Bloodstone Hunt"- My favorite Cap story, granted I haven't read a ton.
Maximum Carnage- I think a 13 part Spider-man story. My favorite Spidey story (once again, there's lots I haven't read.) Also, there's a SNES/Genesis game based on this story, if you're also into old-school gaming.
New Avengers/Civil War/Secret Invasion- New Avengers is a really good read. New Avengers eventually took part in the Civil War (which is its own good 7 comic story line). After this, the team split into two teams and two comics, the newcomer being "Mighty Avengers." New Avengers remained the one to read. After this the story eventually led into the "Secret Invasion" which was also a good series (8 comics). Issues 7-19 of Mighty Avengers also tie in to the Secret Invasion. It's just a lot of comics. New Avengers 1-47 lead up to and are a part of the Secret Invasion. Plus CW (7), SI (

and maybe (14) MA = 76 comics.
Secret Wars- 1983's big Marvel Comics event. 12 issues, gives you a good glimpse into that era. Excellent story.
Marvel Zombies 1 & 2- What would happen if pretty much all the superheroes became zombies? Who would stop them? An amazing story is what would happen. Marvel Zombies 3 and onward are pretty big disappointments after how good the first two were, so don't bother with them.