Hey!
I've programed both DirectX and OpenGL (C# mind you), and what sometechieguy says is very true.
DirectX is much simpler to just 'jump in', you can have a game made in a few days (or hours, whatever your programing skill is). But its slightly less powerful, and more restricted (ie, can only do windows/xbox). One slight problem with DirectX is how everything didn't really connect together (like DirectInput hasn't been updated in years and you're better off writing your own hook).
OpenGL I felt completely lost in at first, but found that thats because a lot of drawn out functions of DirectX are just single commands in OpenGL. OpenGL is much more powerful, and runs on practically anything (windows, linux, ps3, etc). One thing I rather liked about OpenGL was the fact that practically everything is contained in one wrapper (well, for C# at least, like with OpenTK).
(And to nullify a flame war, by "powerful" I mean DX 10 is JUST now coming out with functions that GL has had for years).
I'll mention as well, if you do choose OpenGL, there are a lot more tool kits (like, OpenTK the
OpenGL Took Kit ), making it easier for fresh programmers to get started.
But, once you've chosen one, and both have a sufficient amount of help and guides out there, but try not to mix them. IE, if you choose DirectX, stick with DirectX guides. One uses right handed matrices, the other left; HLSL functions dont match GLSL (mul in particular), and a few other things don't line up for beginners.
Whatever you pick, you can't really go wrong. What helped me out in the end, I started making the exact same game in both DX and GL and found I understood DirectX a lot more (all those years of messing with XNA probably biased me, but hey). And I went with DX, best of luck with your in your decision (and I hope I can help you if need be).
EDIT:
I noticed you said you just wanted to start programing and making games today. If I might suggest, you check out XNA. Its a toolkit made for DirectX that really helps beginners. It handles all the device catching and really lets you just start making a game. Yes, its C#, but the brother of C makes it super simple and fast to learn. (P.S.
Riemers tutorials are amazing).