First, there is no such thing as building your credit quickly (unless somehow you have a time machine where you can speed up time). In credit terms, one month is one month - you can't change that and assuming that you make payments on time, you an get about 10 points added to your score every month with age of accounts and more ontine payments.
As for a credit card, you may have to start with a secured credit card where you put,say $500 in a savings account (and leave it there) and the bank then issues a card with a $500 limit. You use the card, being careful to keep your utiliization under 30% (never more than $150 on the account at any one time) and pay the entire balance off when the bill comes in )never before the billing date and absolutely never after the due date). After a year of on time payments and smart use of the card. you can go back to bank and ask for an unsecured card (which frees up the money in the savings account).