There is nothing in the Bible that says we cannot use doctors. In fact, Luke is commonly believed to have been a physician. So, God has chosen a doctor to help write the New Testament.
God has gifted believers, as well as unbelievers, with the ability to study medicine and to be used by Him in order to heal people. In fact, God has gifted all kinds of people in all kinds of ways. What about going to the orthodontist to have our teeth straightened? As Christians, should we just pray about it and not go? Or what about people with bad vision? Should they pray and ask for healing or go to the ophthalmologist? The answer is simple. Do both. Pray and ask the Lord to heal you. If He doesn't, go to a doctor.
But whatever you do, don't ignore serious medical issues just because some preachers say that people who go to doctors don't have enough faith. If a preacher complains in such a manner, shouldn't he be consistent and also complain that you don't have enough faith to have your car miraculously run when the engine blows up? Or what about your TV, your stereo, or computer? Can God not heal these things as well by His sovereign hand? Of course, He can. But the simple fact is, very often God chooses to use people to make things better.
So, if you are sick, pray and ask the Lord to heal you. Expect healing and don't doubt, but understand that the healing you may get could be through miraculous intervention by God, or the natural healing of your own body, and/or the work of a doctor who might use surgery and/or medicine. Either way, God is the one who miraculously works and who also has gifted people to be able to perform their healing work.