I am the same way (programmer, no design skills), and I found that the perfect match was C# and SFML.net to provide basic graphics/sound/input functionality. I found SFML very intuitive and natural, and while it won't give you the large scale game architecture support that game development libraries might, it gives you what you need, which makes things very easy at the start. And, considering myself a programmer more than a game-developer, I prefer working at the level of abstraction that you get with SFML. Unity was too high level for me because it abstracted away so much of the programming that nothing made sense.
Now start off simple with something like Pong, just to get your feet wet, moving on to things like Space Invaders.
I think a great 3rd/4th project that will really expand your abilities is a shmup (Shoot-Em-Up, think Galaga or something similar). A Shmup will get you into animations, more complicated collision detection (without complicating things too much with complex physics), audio, game states, multiple levels and all kinds of other things if you want to take it that far.
With the early projects though remember that you are trying to learn at this point, so pick new projects for what they can teach you.
