The first thing you need to do when looking for a new bank is to sit down with as many statements from your current bank as you can find. If you bank online, this couldn't be simpler. Then, go through them month by month, looking at your expenses and the fees your bank charges you.
It's important to be honest about your behavior. If you're changing banks as part of a personal program to improve your finances, include that, too -- but now isn't the time for unblemished optimism. Your bank should fit your needs, not the other way around.
Are there fees or policies that annoy you? Some big banks play tricks, like stacking payments, so that you end up paying more in fees than you would elsewhere. Do you pay for things like online banking or bill pay that you don't think you should have to pay for?
Are the services you pay for now coming at a fair price? Are there conveniences you're willing to give up for services that would fit your needs better? You may be able to find a compromise elsewhere, but you won't know for sure until you've looked at what you use on a regular basis.