Once you have recruited Blathers (check out our guide here for how to recruit Blathers), visit him in his tent and speak with him. In order to build a museum, Blathers needs to have a total of 15 different fish, insects and/or assessed fossils. The 5 you previously gave Tom Nook count towards this total. Blathers will give you DIY recipes for a shovel (which you will need to dig up fossils) and a vaulting pole (which you can use to jump across a river). *NOTE: You can purchase Pretty Good Tool Recipes for 3,000 miles from the Nook Stop located in Resident Services using the Redeem Nook Miles option. If you purchase these recipes, your tools will last longer and make it easier to collect items.
Once you have crafted the items, head out on your island to explore new places! The spots marked with cracks on the ground (see image below) are where you can dig up fossils.
Once you have an assortment of different items, return to Blathers to donate them. For the first fossil, you can just give that to Blathers, however, the rest you will need to first have him assess them before he will accept them as donations. When you have donated a total of 15 different things to Blathers, you will have to wait a day in real life for the museum to be ready. While you are waiting, Blathers will not accept any donations, but he will still assess fossils.