You'll have to buy a net, a fishing rod, and a shovel from Tom Nook. Then go out and have at it.
It's obvious how you use the net and the fishing rod. Whenever you see a bug or a fish, just go for it.
You'll find fossils by digging in certain spots in the ground. The ground will look different from normal where there are fossils.
Pictures for the museum come from Crazy Redd's, so be on the look out for him. Be advised though, they aren't cheap.
Go shake the trees around your town, you may find something other than fruit in them. And if you feel angry, or not, feel free to hit rocks with your shovel. You might get something from it.
Get friendly with the citizens in your town and write them a letter with a fruit as a gift. They might just send you something back.
And of course, don't forget every Saturday night at 8, a special guest comes to The Roost. The Roost is located in the bottom of the museum.
Hope that helps some. If you have any other questions, feel free to ask. :wave: