You will need a drill with the appropriate drill bit (1/8" bit for wood or a 3/16" masonry bit for stucco or brick), a screw driver, a hammer and maybe a utility knife.
Doorbells only use 9 or 14 volts of electricity, so you don't need to turn off the electricity.
First remove the old doorbell. If it is just a button, pry it out with a screwdriver. It is just held in by friction. If it is screwed on to the wall, remove the screws.
Detach the wire leads from the posts on the back of the button by loosening the small screws on either side.
Remove the old doorbell.
To mark the holes on the wall for your new doorbell surround, hold it up aligned with the old hole in the wall where the wires come out. Make sure it will fit back into the hole. If it won't use a utility knife to remove enough material so it will fit. Make sure you have it straight on the wall and mark the holes with a pencil or awl.
Using the appropriate bit (You really do need a masonry bit to drill into stucco or brick!), drill where you have marked the holes.
Pound in the plastic anchors for stucco or brick.
Attach a wire lead to each post on the back of the button. It does not matter which wire goes to which lead.
Put the button-back into the hole along with the excess wire and line up pilot holes you drilled for the wood screws or the anchors you put in for the stucco and screw in the screws. DO NOT OVER-TIGHTEN THE SCREWS!