Answers

You definitely want to make sure the person is licensed and that they obtain the proper permits for this.  Most "discount" electric panel change-outs are not done well or properly.  Proper licensing and permitting will protect you.  Try to be home when the inspector comes or contact your municipality directly to verify the inspection was done.  Do this BEFORE paying in full.  

Quality craftmanship, good service and low price do not happen very often, if at all. You have to give up something if you don't want to spend much. As Rob said, you don't want to cut corners when it comes to electrical work. Have them trace what devices are on what circuits. Putting in a new panel with the same size breakers (that are overlaoded) isn't going to solve your problem.

Being that this is a saftey first and homeowner's insurance second question, a Licensed Electrician in your county/township is the best way to go. Breakers tripping can be one of many issues from, just a faulty breaker, to improperly installed or overloaded circuits. NEC is nationwide but county codes/variances can be more stringent/specific in residential electrical. Electricians are not cheap but this is an area you don't want to take any shortcuts. Ensure you see their license and make sure the work gets inspected prior to energizing your home.