The first thing to know when removing a one piece tub surround unit from your existing bathroom is that you are going to separate glue and sealant from the tub and wall. It is likely that all the wall panels and the tub edges, as well as any trimming, are held with a water tight sealant. You are going to need to clean all the sealant up before installing any new fixtures. Exert caution while doing this process to limit the amount of mess, and to avoid damaging your walls.
Step 1 - Remove all Fixtures
The first thing to remove are the actual fixtures in the one piece tub surround unit, so that once you have it freed from its position, can simply pull out of its resting place. Use a screwdriver to remove the securing screws on the handles, and then pop them off. Set these to the side if you are going to reuse them in a new unit that is being installed. Try to keep all parts and screws together, to avoid them from getting lost. The actual faucet will generally screw out on a counter thread, meaning it's not left to loosen and right to tighten, but the opposite, which prevents it from simply coming loose. Use a wrench to take this off and set it to the side as well. This would also involve removing the drain, and you can stuff a rag into it, to prevent waste from falling into the line while you work.
Step 2 - Remove the Wall Board Trim
Use your hammer above the splash guards or wall panels around the edge of your one piece tub surround to remove the trim. Once these are removed you should see a gap between this trim and the wall, and there should be sealant in there as well. Use your putty knife or even the flat razor to cut the sealant and release the panels from the wall without damaging it. Once you have these removed, you will then see the same gap between your tub and the wall. Use the flat razor to shave the excess glue and sealant from the walls so that they are back to being a flat surface again. This will prepare the area for future installation.
Step 3 - Remove the Tub from Position
Once you have the faucet and the walls removed from your one piece tub surround system, you will follow the same basic steps to remove the tub from its position. Some units have rear or forward access holes, where the tub basin could be bolted into place, which is very uncommon. If you do have this kind of tub, it will involve unbolting the unit before trying to take it out, to avoid ripping chunks from your floor. Take the flat razor to the sealant around its edges and use your pry bar on its edge to pull it free. You can save your wall from damage while doing this, by placing a long flat board between your pry bar and the wall, which will equally distribute the torque pressure from punching a crowbar hole into your wall. Doing this will also help provide added extra leverage.