IF you're interested in experiencing the games chronologically, start with Black Mesa, a Half-Life 1 mod released last year which is basically an HD remake of that game (that aged terribly) with a lot of fixes (highly acclaimed), then play Half-Life 2, then the two HL2 sequels.
If you just wanna have fun, I'd say skip the first game altogether and start with HL2 since it's stellar and amazingly long.
I think a lot of people would say jump into Black Mesa. Changes a lot of Half Life but I think most people would agree that the changes are for the better.
I haven't played it, but I've heard some terrible things about balance and difficulty in that game, alongside stability issues and other things.
On-topic: There are some references in the second game to things that happened in the first game, but the second game does not depend on the first one. You should play both, but you can play them in either order. I played the second one before I played the second one first and it's a better designed game, but the first one has more enemy variety and more fun gunplay in my opinion. It's just that the finale, the Xen level, is atrocious - you should be aware of that.
Also, Half-Life Source is a fine way to play Half-Life. It looks a little better and supports higher resolutions, but don't think that the "source" part means it looks a whole lot better. It doesn't. It just looks smoother and cleaner.
I personally liked the original engine more than Source. I'd just play them like they were meant to be played and play the original first and then 2. I still am yet to play Half-Life: Source but I have played Black Mesa. Even though it's an old engine version of an even older game, I still thought Black Mesa was beautiful.
If you're typically alright with playing older games, skip Black Mesa and just play HL1/source (doesn't matter really). Black Mesa is interesting and all, but it also cuts out some content and is missing the last portion of the game.
Black Mesa is OKAY, but honestly, the original deserves its due. It's one of my most favorite games of all time, and as long as the graphics don't make you violently ill, I think it's worth playing over BM.
Black Mesa is probably the one you'd want to start with. That was such an excellent version of the original half life. If you're really curious about it go through the original game and the source engine one just for comparison sake. But definitely play the original first! You don't need to do it but there are winks and nods to the original game that might be more meaningful to you if you went through the first one.
Half-life is a great linear FPS with little story (how I remember it) and few "cutscenes" (pretty slow start) and it is marred by platforming and the ending.
Half-life 2 is a "rollercoaster" that is highly dependent on the physics "gimmicks" and driving sequences. It has a great ending and it is probably the best post-2000 PC single player FPS experience ever (along with Far Cry and Bioshock). I'm not sure if we'll ever see a single player campaign like that again (CoD doesn't count).
So if you hate challenging shooting and/or platforming skip Half-life but don't miss HL2 (the episodes are kind of forgettable).