The best way to get around within Uddevalla is to walk, cor cycle during the summer months. However, if its winter, I suggest taking the bus.
In the summer one can also take the boat to different beach locations in Uddevalla. You can also take some of the busses to these locations as well.
If you want to go to the large shopping mall, known as Torp. , then I suggest either to go by car or bus. Only because its some distance between Torp and downtown Uddevalla and I think walking might be outta the question.
Around Uddevalla Centrum:
Its very convenient to go around by bus.
This is at Kungsgaten or the Central.