Well, you could try, of course, but you might have a trouble getting approved for a reward credit card. They are considered "elite products" (justifiably or not is another story). Check your credit score with Credit Karma or/and Credit Sesame first. See if it's close or over 700. If not, start with just a regular credit card with no annual fee.

If you still want to try, I'd recommend Chase Sapphire. The 40,000 points bonus can be transferred to many programs; it fits your spending pattern; and I think Chase might be more liberal toward new members than Citi and Amex. Be ready for the initial denial and having to call back and making your case. Check Chase reconsideration thread in this forum.

If that doesn't work, just get some regular credit cards from all major banks and in about a year you'll have "conditioned" them to the point when you're able to apply for most of the cards you want. Those bonuses are not going away, at least not that quickly.

Finally, you might already know that, but (especially with reward cards), do not carry a balance ever! Make sure to pay off all your balances every month unfailingly and on time, or you'll find yourself in a world of trouble. Another words, do not treat your credit card as a personal loan dispenser and you will be fine. Good luck!