The tub/shower unit seems to be made of acrylic, has no lines and looks like it is only one piece. I would like to replace it with a newer version with higher walls. Is this possible and if so how?

Remove an old plastic bathtub and surround and install a new bathtub and three piece surround with new plumbing and a tile ...

the Tub-to-Shower Conversion Kits can be customized to fit nearly all shower enclosure sizes and conditions much more quickly and affordably than rigid one-piece acrylic or composite-shower pans.

Beware of one piece bathtub shower surrounds and one-piece tub/shower stalls. They're usually too bulky to get into an existing bathroom. Also pick up a new drain and overflow assembly ...