The museum is where you collect the various items you can find in your world.

Aquarium
There is an aquarium area where you drop off one - and only one - of each fish. Catch them with your pole, then talk to the owl in the museum. He will take one of each fish to put into the tanks for you. You can go admire them later.

Fossil
There's a fossil area where you put the fossils you dig up. You need to wait until you get a shovel to do this.

Observatory
There's an observatory upstairs, where you can look at and even name constellations. You can be told which day will allow you to see each one you've named.

The Club
Downstairs, every Saturday night a dog will come and sing songs. If you know the name of a song you can request it from him - if not he'll play a random song for you. Once he plays a song all the way through, you'll get a copy of it to take home and use on your radio system.

Animal Crossing Wild World Walkthrough