Do you bank? Your bank would likely have a credit card they can issue you. If you're a college student, there should be plenty of available student cards. I have a Citi one.
If you've had a checking account for a while, you should already have some credit established (good credit, assuming you haven't overdrawn or bounced checks). Stick with a well known insitution (Citi, Providian, etc).
You can also get CCs from stores, like Best Buy, Sears, Amazon, even PayPal (I have one).
As for different types, I don't know that you'd have much choice right now. Visa and MC are interchangable, but I wouldn't go with a Discover or American Express for a first card. AE has some very different terms, and Discover isn't used everywhere. With a first card, it's unlikely you'd have things like cash-back, FF miles, etc.
Some other tips:
Get a card with a low minimum finance charge. Usually these are between $.50 and $1.50. Get the lowest you can. I know you're saying to yourself "but I'm never going to charge $.50 to a card", but you never know. I've seen them as high as $2.50. This means a 50 cent can of soda will cost you $2.50,
Stick with fixed rates.
Don't be fooled by offers that say "as low as" a certain percent, because it's almost gauranteed that you won't get that rate.
CCs are almost essential to have for emergencies, but also have emergency funds in your savings/checking account. That way, if you need a quick $300 cash (actual cash, not a purchase), you won't be charged 24% for an advance.
Once you start using the credit card, you'll get flooded with offers for more. Don't ignore them. Look for one that has really good rates and terms. Find the best, and send off for that one too.
But, when you get the better card, don't cancel your other one. Having long term accounts benifits your credit rating, even if you don't use it.
Finally, even if your card has in introductory "6 months no interest" thingy or something like it, pay it off quickly anyway. It can add up quick.