A fico score over 800 should qualify you for the best interest rates. Lenders also look at loan to value, debt to income, location of the home-suburban/rural and size of the loan when they determine interest rates. Call up to three mortgage companies, banks or credit unions and compare the interest rates and closing costs that they are currently offering you. They cannot lock your interest rate in until you have submitted a contract but they can give you a close estimate at your interest rate and payment.
Its hard to determine how low your closing costs will be without knowing many factors. Get a few referrals of mortgage companies and compare them.