To get an overview of Wolverine, there are a number of storylines that can be read to give a nice overview of the character. Wolverine is a character that has been around since 1974, so he has a lot of history, both with and without the X-Men. I'd recommend:
Incredible Hulk #180-181 (1974): Wolverine's debut appearance from 1974. It's his first appearance, so it's as good a place as any to start.
Giant-Size X-Men #1, X-Men #94-100 (1975-1976): Wolverine joins the X-Men as part of their second generation (alongside Colossus, Storm, Nightcrawler, etc.). This collection covers his first adventures with the team.
X-Men #109, 120-121 (1978-1979): Wolverine's origin is briefly explored for the first time, mostly covering his history with the Canadian superteam Alpha Flight.
Wolverine #1-4, Uncanny X-Men #172-173 (1982): This is the first solo Wolverine story, covering his time in Japan and courtship of Mariko Yashida. The issues of Uncanny X-Men cover his attempted wedding with Mariko. All are pretty integral to his character, and this is widely considered to be the definitive Wolverine story.
Uncanny X-Men #205, Marvel Comics Presents #72-84 (1986, 1991): Barry Windsor-Smith told some definitive Wolverine stories, including the epic Weapon X storyline that showed how Wolverine got his adamantium skeleton. This is also considered essential.
Wolverine: Origin #1-6 (2001-2002): After nearly 30 years of existence, Marvel finally got around to telling Wolverine's definitive origin with this mini-series. This goes all the way back to Canada in the mid-1800s and starts with Wolverine as a child.
X-Men: Schism (2011): If you want to jump into the current adventures of Wolverine and/or the X-Men, then this storyline is essential since it sets up the current status quo.
Wolverine & The X-Men #1-7 (2011-2012): Wolverine currently runs a school for mutants in Westchester, alongside the X-Men, and this is where that storyline starts.
As for your second question, yes, Wolverine and X-Men comics are still being published today. Comic books as a medium tend to latch on to what is popular and publish them until they are not popular anymore...or even well past it. That's why characters like Superman, Batman, Spider-Man, the X-Men and Wolverine have been continuously published for the past 40-70 years. Just this month alone, March 2012, Marvel is publishing 16 comics starring either Wolverine or the X-Men
- Avengers vs. X-Men
- Ultimate Comics X-Men
- Astonishing X-Men
- Uncanny X-Men
- Wolverine and the X-Men: Alpha & Omega
- Wolverine and the X-Men
- Uncanny X-Force
- X-Men Legacy
- Daken: Dark Wolverine
- New Mutants
- Generation Hope
So there's a lot to choose from, and if you go into a local comic book shop, you'll see Wolverine on the covers of a bunch of comics. I'd recommend starting with any of the ones I've linked to.