You should be very careful choosing any financial institution.
Weather you're looking at getting a new checking account, or savings account, or credit card, do your homework. Read reviews, go into a branch and ask them about their online presence. Ask your friends about their experiences, and ASK ABOUT THE FEES.
Most banks charge monthly service fees. MOST. Not all. So there is absolutely no reason you should every pay a monthly fee to let the bank use YOUR money. You are doing THEM a favor, not the other way around. For Credit Cards, well... you're going to pay interest, regardless, but you do NOT need that special rewards card that charges you $50/year unless you're REALLY going to use it. Remember, if it didn't screw their customers and make them money, the CC in question would charge more.
Look at local banks. Many larger banks died in this recession, but a lot of smaller local banks, who invested in their communities and actually paid attention to what they were investing in are doing just fine. They frequently have much stronger customer service than large banks, but also usually have a much weaker online presence. I would also suggest, if you choose a small local bank, looking very closely at how they're guarding your online banking information. Unfortunately, small banks tend to have great customer service with not great security training. I have been able to talk two of our local banks out of our client's banking information with very little in the way of credentials, and a whole lot of friendliness. This has been convenient, but not very reassuring, so put them through the test before you decide to put all your money with them.
Look at Credit Unions. I have never had an actual bank account with a real bank, I have always had a Credit Union. They have many of the same benefits of a regular bank, with the added bonus of generally having better interest rates, because they're not built to make a profit for some nebulous head honcho, they're built to make a profit for their members, meaning YOU. They tend to have good customer service, and varying online presences, but be aware: they generally have really inconvenient hours, and ATMs can be challenging to find. Most Credit Unions will be part of a network, though, and you'll be able to use any ATM in that network, free. Occasionally this means that deposits you make in an ATM can take a while to go through, so be aware of that.
If you choose to go for a major national bank:
Find an account that charges no monthly fees, no matter what. They have them, and you deserve them.
Don't let them sign you up for any sort of banking software, or billpay software, unless it's totally free. It frequently doesn't actually DO anything, and costs $20 a month that you don't need to spend.
Shop around for high-interest savings accounts. ING Direct has gotten good reviews. Be aware, though, that you'll never find a savings account that pays higher interest than your CCs charge, so pay those off before you really start throwing a lot of money at savings.
Look for a bank where you can have a "Personal Banker". This is someone who is basically YOUR customer service rep. They will always be the person you contact when you need help, instead of some random customer service rep in a phone farm on the other side of the world. After you've found one, make an appointment, and get to know them.
In summary: Big Banks tend to be evil about fees, and terrible about customer service. Small local banks have less fees and great customer service, but not great online presence. Don't forget to look into Credit Unions.
Think about how you use your bank, and how you want to be able to, and then shop around for something that works for you. And remember, you're doing THEM the favor, and you should not have to pay them for the privilege.