You can easily upgrade your shower in a weekend with a new fiberglass shower base. The shower bases are made to drop in over the existing flooring and provide a contained water tight base.
When considering remodeling your bathroom for a new shower, you should look at the current configuration and internal structure. For example, does your subfloor have a proper drain system in place? If that is the case, you can literally install the new shower base directly on the floor.
Your shower base will go over the existing drain and the be sealed to minimize leaks. You can then install walls or tiling to complete the shower. Typically, you can purchase glass doors for the shower entrance that are quite simple to install. The most time consuming part of the project will be the tear out of previous fixtures and securing the shower bases bottom to the floor.
To secure the shower bases to the floor, requires preparing a level spot. You will want to insure that the shower bases drain correctly into the shower drain to limit any puddling. That might require shimming the shower bases from underneath. Once you have prepared a level spot, you can use construction adhesive to lock in the shower bases underside to the floor.
Regardless of the shape and style, if you start from a good level base you will have a more enjoyable shower experience when you install your custom walls or tiling too.