Start with your current lender. If you're a good customer-you hold a sizable mortgage, pay on time, and maintain good credit-your existing lender will probably do everything in its power to keep your business. The company may cut you a break on fees for things like appraisals, surveys, and inspections if the information is current and you meet other requirements.
Don't fall for the 0% apr unless it fits in with your master plan. A lot of brokers will try to get you locked into a low interest rate that will balloon on you in a couple of years and leave you out on the street.
If you are planning to stay in your home for several years, then you can save money by paying points for lower interest rates. You pay up front fees to ensure you have lower interest payments over the course of your loan. Remember, this only works if you keep your mortgage for several months.
Make sure that your original mortgage does not have a pre-payment penalty or early payoff penalty of any kind. Sometimes people will get into their mortgage with the mortgage having a pre-payment penalty and they will not even know about it. Pre-payment penalties usually range from 6 months to 3 years with a penalty for an early payoff. The penalty is usually about the amount of 6 months worth of your mortgage loan interest, but this varies. You would have to be able to have some significant payment and interest savings on your refinance loan to justify refinancing a mortgage loan with a pre-payment penalty.