Wait until the next morning
Your island "resets" at 5 a.m. each day. What this means is that if someone tells you that a character or building will arrive "tomorrow," you'll have to wait until 5 a.m. the next day.
Once 5AM hits, you can boot up the game and head to the spot that you placed Blathers' tent. Go inside.
Talk to Blathers
Once you're inside the tent, talk to the owl who's there. This is Blathers, the museum curator, and he's looking to start a museum right here on your island!
There's one issue, though: he doesn't have enough specimens to fill an exhibit, let alone a whole museum. As such, Blathers is tasking you with finding him enough bugs, fish, and fossils to fill the building.
To assist with this, Blathers gives you the DIY recipe for the Vaulting Pole, which will help you cross the rivers.
But this is just the recipe — to actually get the Vaulting Pole, you'll need to craft it.
Craft the vaulting pole
Use the vaulting pole recipe in your inventory to "Learn" it, so it'll be added to your list of DIY recipes.
Now, you'll need five pieces of softwood. You get softwood by hitting trees with an axe. If you don't have an axe, you can craft one with five tree branches and one stone.
When you hit a tree with your axe, a piece of wood will pop out. You can hit any given tree three times before it'll stop giving wood.
There are also three different types of wood, which will drop (mostly) randomly: wood, hardwood, and softwood. You'll want the last kind, which is lighter colored than the rest.
Once you have your five softwood, head back to the crafting bench and build your Vaulting Pole! It won't ever break, so don't worry about crafting more than one.