Really two questions here:
1. Should you get a credit card?
2. Which card should you get?

It's never too early to start building a good credit history. It's almost always too early to start building credit card debt. As noted above, you do not want to carry a balance. The interest charges add up very quickly. If you have the discipline to only charge what you have, and to pay the bill, in full, every month, then get a card. Besides building the credit history and earning rewards, having a credit card also makes it easier to rent a car or stay in a hotel.

If you do decide to get a card, look at interest and fees as well as types and amounts of rewards. The interest rate should be irrelevant, since if you pay in full every month then you'll never pay interest. (But if you will carry a balance and are ignoring the advice not to, then getting the lowest interest rate should be your primary/only concern).

Fees are important also. The majority of credit cards out there have no annual fee, but that's not true for mileage cards. The AA card fees run $50-85 per year or more. If you only charge a few hundred dollars per month, you won't be earning enough miles to make up for the fees you are paying. For example, $400/month would earn about 5K miles per year, it would take 5 years to earn that basic free domestic roundtrip ticket. So you'd be paying $250-425 in annual fees to earn a free ticket that you could probably just buy for that same money. For this amount of spend, you're better off with a card that just earns cash back.

If you're spending more, have the discipline to pay the card off every month, and especially if you have your eye on high end tickets (international business or first class), then the AA card may be a good deal.