First, is this a blued, nickel, or parkerized 1903?
I'd recommend using just plain 3 in 1 light machine oil or CLP. Use some flannel 12 bore patches (large white gun cleaning patches)
If your going to treat rust on the frame, remove the grips. (if they are the hard rubber / plastic grips, be careful not to over-tighten them, they can crack when shooting if too tightly screwed back down). If you are just treating the slide, remove the slide. I'll assume that you already know how to do this. It is an adventure the first time, so if you don't know, PM me and I'll walk you through it.
Apply oil liberally in affected area. Let soak overnight.
Remove as much oil as possible with patch. Apply more oil to patch's rough side. Rub until your thumbs are sore. Repeat this process for a few nights until most of the rust is gone. Use a slightly less amount of oil each successive evening. You should see rust on the patch almost immediately. Use a few different patches each night. Rough side to remove rust, smooth side to apply oil.
I've seen some NASTY looking rust on old Colt's clean up to the point that there would just be a few small localized spots of stubbron rust that won't leave no matter what by using this method. If the gun is parkerized, don't rub as much. Soak more and rub less. If it is a nickel gun, then I cannot help you, but oil will stop rust in the majority of cases.
outside of that, best of luck. Take some before/after pics and post your progress up here. The 1903 is a slick little gun. VERY pointable and quick. I used to own one, but I bought one in unsatisfactory condition and sold it. I will buy another if a good price on a good "shooter" grade comes up again.