Every time you apply for a credit card you provide the issuer with personal information. Credit issuers always check this information before they decide whether to qualify you or not for a credit card. Also, issuers request your credit report to see you credit performance and estimate your creditworthiness. If you apply several times in a row for a credit card you may hurt your credit score and issuers may deny your credit card application for too many inquires. So let some time pass before you apply for a card again.
You should use a secured credit card for at least a year before you can replace it with an unsecured credit card. But to do this you should use your secured credit card wisely, (i.e. make on time regular payments and pay your balance in full every month.) Also, this credit performance will allow you to improve your credit score. In about six months of perfect credit behavior you will see your credit score go up. You will notice the first signs of credit improvement after two or three months of responsible credit card use.
As for student credit cards, you should consider Discover student credit cards (see second table with «Student Credit Cards»). Some of these cards have no annual fee and offer rewards (e.g. Discover® Student More Card – $20 Cashback Bonus).