The Pewter Museum is located in the far north of Pewter City. It will cost you 50 Pokémon Dollars to get into the guest part of the museum. On the other side (The "employees only" side), you are able to receive an Old Amber from the scientist, but the only way to get to that side is to Cut the small tree.
However, during Generation II, the museum is closed for renovations.
During Generation IV, the museum was open, but only reduced to minor exhibits and one room. Fossils could also be revived here.
The Pewter city Gym is located on the west side of the city. The Gym leader is Brock and he uses Rock Pokémon. After you win, he will give you the Badge and you will be free to leave the city. Brock comes off as an arrogant trainer wanting to see if the challenger (you) is really good enough.
In the Anime
In the Anime, the camera focuses on the Gym, not the rest of the city.