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Most credit cards are airline-specific, with your points going directly into your account with that airline. Amex (the regular non-co-branded one) is an exception, if you join their Membership Rewards program, the points can later be transferred to a variety of programs. It does have a fee, on the high side.

A good alternative may be the SPG Amex, which earns Starwood hotel points. These can be transferred to many airlines, at 1:1, or 1:1.25 if you move 20K at once. You may find that the points are even more valuable for hotel stays. And the fee ($35?) is lower than many airline cards. It's a favorite of many of us here. Having the card gets you Starwood "Preferred" status, and if you spend $30K/year, you get Gold status.

I'm not aware of any airline credit cards here in the US that give you airline status just for having the card.