Only thing is that Chase Sapphire Preferred, Ink Bold, Ink Plus, or J.P. Morgan Select are the only cards that gives you 1 point to 1 mile transfer to MileagePlus. They transfer to British Airways as well. And since you don't currently have a credit card (although you have some credit) it may be hard to obtain those cards. Although it wouldn't hurt to try.

If you can't get one, get a card that participates with Ultimate Rewards, that way you can build up points (which never expire), and when the time is right, apply to one of those cards listed above and transfer your points there. The Chase Freedom and Chase Sapphire (not Preferred) can be used until you get one of the cards that transfers to miles.

Feel free to PM me if you have any questions, the whole process may seem very confusing, I know I was.