So much to share(!) We have been to Disney Vero Beach at least 10 times and love it. It's a summer home vacation for us.
Above and beyond the resort itself, some of our favorite things to do in the area include going to the beach at Sebastian Inlet State Park. The Disney Vero Beach beach is great, but the Sebastien Inlet beach is stellar. There's a small fee to enter the State Park, but it's well worth it. Hardly anyone there in the summer and really beautiful water and waves if you like to skim or boogie board or surf (near the pier for that). We also like to do the Florida Cracker alligator airboat once in awhile (super fun and not too far), plus going to see the manatee at Round Island Beach Park is a good (and short) drive. Walking around Vero Beach itself is fun as well to check out a few local shops.
For dining, some our favorite stops beyond the hotel include Orchid Island Pizza (a quick walk from the hotel and we like their pizza and other items), Squid Lips in Sebastien (especially for the ambiance in its over-the-water location), Ay Jalisco for Mexican in Sebastian, Ocean Grill in Vero Beach for a nicer upscale meal at the beach, and Casey's Place for a local legend casual outdoor burger joint in Vero Beach. These are just a few quick things off the top of my head, but rest assured there's lots to do in the area (not to mention all the fun stuff at the resort itself). Just beware of the noseeums(!) Such a great spot in a great area.