When searching for a new credit card you have a couple options to take. You could apply with a small bank or a major credit card company. Before you get into a application you need to think about a couple things. Do some research and pick at least three different companies and choose from them first.
Do you have already have accounts like a car loan, mortgage, or checking account with a bank already? Then keep it all in the same house. This way it makes it easier to keep all of your finances with one bank instead of a bunch of different ones. Transferring money between accounts to pay bills is and view balances is much easier now. Calling customer service is much easier because now they have all of your info in front of them making it easier to answer your questions.
Most people have accounts with different banks already. Small banks will value you as a customer because they need you. Rates at small companies might be higher but a quick call to customer service can probably change that. They cannot afford to lose you. You can expect quicker customer service turn around times at a small bank.
If it is options that you want then you will want to apply with a major card company. Some big banks have so many credit cards it might take you an hour to choose from all of them. You should be able to narrow it down eventually. The customer service might not be as good as a smaller bank but you can get what you want as they are competing against other large banks for your business.
Going with a big or small credit card companies will not matter if you take the time needed to find the right card. Do not rush into a credit card application without thinking if the rewards program or company you are working with is going to be able to take care of you into the future.