You can replace your old tub spout and improve the look of your tub. Be careful when choosing your new spout: some spouts have a little pop-up knob on them for turning on the shower; others have a sort of dial. If you do not replace yours with a similar type you will need to adjust your plumbing, or you won't be able to turn the shower on and off.
You will not have to turn off the water to the house because water only comes out when the tub knobs are turned on. Usually spouts are placed hand tight. The older the tub the tighter they will be. The first test is to see whether you can turn it with your hands. You can place a small face towel around the spout to avoid injury to your skin. It should turn counter clockwise; do not force it.
If it will not turn, check to see if the spout is caulked to the tub or wall. If it is, you will need to use a small sharp instrument to remove the caulk. Proceed with caution to prevent injury or damage. If you plan to reuse the same spout and have to use a wrench to take it off, you'll need to protect it with a rag so the surface is not damaged. Gently pull the wrench and see if it loosens up. Do not use a lot of force. If it still will not come loose, tap the wrench with a hammer in the direction you want the spout to go. Use slight taps, like you would use to ring a bell. By now it should start to move.
Take the wrench off and try to remove the spout the rest of the way by hand. Do not continue to use the wrench because you could be twisting the inside pipe and it could break. Alternate these methods until the spout can be easily removed.
When the spout comes all the way off there should be a small copper, steel or plastic pipe sticking out of the wall. It will have a threaded end. Gently clean the threads with steel wool. Clean the tub or wall where the caulk was. Place Teflon tape or pipe joint compound on the threads coming out of the wall. Screw the new spout on by hand until it is lined up. If it will not start easily, clean the threads some more.
Check the operation of the shower knob if applicable. Now your tub spout should look nicer and work better.