Stripping a screw is indeed a terrible thing to do. It is best to avoid the problem of stripped screws by not stripping them in the first place, but you should also be prepared to deal with a stripped or rusted screw in the event that you have to deal with one.
Your first step should be to look at the screw's head and find several screws that you might think fit well in it. Don't bring one and try to force it to work. Bring several and test as best you can to find the one with the best fit.
Take your time. When driving a machine screw into a tapped hole, line it up and start driving it, being gentle and operating on the assumption that the screw will go in without much effort. Again, do not "force it" to work. If effort is required, assume that you have crossthreaded the screw and take it back out. If, after several attempts, you have determined that you're not crossthreading the screw, try to figure out why so much force is being required. If the tapped hole is large enough, you may want to try spraying it out with compressed air. You can try putting a drop of lubricant, a tiny spray of WD40 and letting it sit for a few minutes, or using anti-seize (lubricating compounds that prevent galling.) If all else fails, you can try driving it home with force (the ability of the driving bit to stay in the screw is dependent on its fit, forward pressure, and torque; in order to apply a lot of torque, you must first firmly press the driving bit forward, into the screw)... or you can drill the hole out a tiny bit and retap it, using machine screws from a local hardware store that you know will fit in your newly tapped hole.
If you are stripping larger (1.5"-3.5") screws that are being driven into wood, you can use an impact driver if you think the screw can withstand the force. (18V right-angle impact drivers have around 500 in-lbs of force for screws that can't take the 1300-1400 in-lbs of force that a standard impact driver puts out.) If you know the driving bit is about to disengage and chew up the screw, back the screw out an inch and try again. The first pass serves as a sort of "ghetto" pilot hole. If all else fails, drill a proper pilot hole. You may need to secure the two pieces of wood together somehow to keep them stationary during the process of drilling the pilot hole and then following it up with the screw. Possible options to secure the work pieces include C-clamps, temporary screws, temporary nails, vise grips, temporary supports, etc.
If you strip a screw, take it out and replace it. (Hopefully it isn't hard to find a replacement.) Removal options include (1) gouging a slot with a dremel tool and using a slotted screw driver to extract the screw, (2) using Eazy-Outs / Alden's "GrabIt!" / other screw extraction kits, and (3) cutting/grinding the head off the screw and then drilling the hole out and retapping it, regardless of whether or not there's a screw still left in the hole. Let the drill bit chew it up.