Which X-Com has a good balance between modernity and awesomeness? I don't really want to play some old old X-Com, but the newer ones might not have as good gameplay. So which is the best of the X-Com?

You may not like the answer, but the truth is, "the first one".
Nothing really compares. Terror from the Deep is also great, but all it really does is add a new setting and a few new features to the first one.
Apocalypse I never got too far in. It's not BAD, but it definitely doesn't compare to the first one.
Look past the "pretty graphics" aspect, because X-Com is totally worth it.
Besides, even the most recent one (Apocalypse) was released in 1997 - 4 years after the original - so it's not exactly a graphical powerhouse.
Since you'll be dealing with old school graphics anyway, you may as well just play the first one and embrace it in all its glory.

You can get all these games, running just fine, with Steam.
In case you're interested:
http://store.steampowered.com/search/?term=xcom
edit: i guess urls don't work.

True. 1997 graphics = fail.
Alright, I will just go with what you recommend.

Someone has made remade the original game with better graphics and I think some other improvments. [url=]http://ufoai.sourceforge.net/[/url]

Reviews for any beyond the second game were anywhere from very mixed to pans. The first two are just much better, and the first one is notably better than the second. The settings are more varied and easy to identify with, and the game is scarier. Additionally, the second one has some maps that are perhaps too big, and are more difficult to see in. Also, there is a monster or two there that are significantly harder to handle than the monsters in X-COM. But not in a fun way.
Go with the first one. It really is as close to perfect as a game is likely to be. There's a reason it always gets voted as the best or near best game so many years later.

As a sidenote: In my opinion X-Com: UFO Defense / UFO: Enemy Unknown is a very pretty game. Graphics-wise. I'm not a "nostalgia freak", but I think the game's graphics are really nice to look at for how very detailed they are. Exploring a lush farm with some moody lighting or fighting in houses authentically stuffed with pieces of furniture and other objects, it just all looks very good.
Obviously that's just my personal opinion, but I felt inclined to say that "modernity" definitely doesn't automatically mean "prettier", no matter how advanced the technology is :)

Apparently, it's not a remake, but an "inspired by X-Com" work. Either way, it looks awesome. I think I will wait on the original X-Com a little bit.

Well, that's your call obviously, but I would still recommend the original.
I think this is still pretty clearly a work-in-progress rather than a full-fledged game. And they will be hard pressed to match what is simply one of the greatest games of all time.
Also, it may sound like a small thing, the fact environments aren't completely destroyable in the remake is a big deal. One of the great things about X-Com is that nearly everything is destroyable given enough firepower, which adds a lot of potential strategies.
Not dissing the remake, mind you. I may just download it myself, and I always appreciate when people put a lot of love and hard work into something like this.
But it'd still be a shame for you to miss one the original. Nothing is quite the same.

So far I'm not really getting that "X-Com feeling" from UFO:AI.
Not really surprising. I have significant doubts a fan project could ever live up to the name, sadly.

Bleh, ok. I'll play the original as I wait for the fan project to fully develop.

I have just acquired the original big box X-Com: Apocalypse. As I am running XP, can I install it then add it to my Steam account to run it?

not in the way you are imagining it. (you can't enter the cd key and get the game on steam)
however you should be able to add it as "non-steam" game.
Nonetheless you need to be creative a bit with dosbox.

See, therein lies the problem... "some other improvements".... X-com doesn't NEED improvements, yet every fan attempt at updating always has some misguided attempt at improving upon the original when all it really needs is some code tweaking to run on modern CPU's and perhaps some higher resolution support. That's it. Nothing more. No need for "modern" graphics, that would throw off its charm.

I for one, like Apocalypse better (not to say the other are bad, cause they aren't).
I just loved destroying buildings in Apocalypse and manage the relations between the several corporations in the city, along the fact that you can mix and match armours and better customize vehicles.
Playing Apocalypse in super human was fun... getting bomber UFOs by day 4 always spices up things :D