That depends. Assuming you're gainfully employed, but you're young and don't have a long history of credit -- you don't own a house and haven't paid off a car -- then your best bet is to look for a credit card that is aimed at people with a limited credit history.
If you are older and are making mortgage and auto payments, and you're hardly ever late with a payment, then you probably have good to excellent credit even without having a credit card. And so I'd be optimistic and would apply for a really good credit card. Do you plan to pay off your balance every month? If so, you may value the rewards points that come with some credit cards. Want a lower interest rate to spread out purchases over time? These are some of the questions to ask yourself while looking for a credit card.
As for not knowing your credit score, if you're intensely curious, you can pay for your credit score (myfico.com will give you instant online access for about $20), but I personally wouldn't (I'm cheap that way). Rule of thumb if you do get your credit score: If your score is 700 and above, credit card issuers are probably going to be happy to welcome you aboard; if your credit score is in the 500's, not so much.
I wouldn't pay for a credit score, because you can probably make an educated guess of where you fall by looking at your credit report, which you can see for free at AnnualCreditReport.com, where you can get one free report a year from each of the three major credit bureaus - Experian, Equifax and Transunion. (Just don't confuse AnnualCreditReport.com with other similar-sounding websites, which ultimately are trying to sign you up for their credit monitoring service.)
It's always helpful to see what credit bureaus have down in their reports about your financial history. Statistics show that 3 out of 4 people have an error of some kind. Those errors can be trivial, or can be serious enough to weigh down your credit score.