Although I have never stayed at Vero Beach, my in-laws live within one mile of the property on the barrier island where the resort is located. I spend a couple of weeks every year in close proximity to the resort and am pretty familiar with everything to see and do off-property.
There is very little else within walking distance of the property to see and do-it is located in what is essentially a residential area full of old rich people.
A few miles to the North is Sebastian River State Park. Never been there.
The "Environmental Learning Center" is a short distance away, located on the intracoastal waterway. A complete waste of time.
The town of Sebastian is actually as close to the resort as the town of Vero Beach. Sebastian is a little seedy-they have been working on the riverfront area and it is much improved, but you are not going to do any sightseeing there. You can take boat cruises from there if you like.
The section of Vero Beach that is located on the island (driving South on A1A from the resort) is a small upscale enclave of restaurants and shops. June will be off-season there (many residents do not stay all year,going North around Easter and not coming back until October) so restaurants should not be crowded. We like to eat at the following-
Squid Lips (overwater grill in Sebastian-the Cajun Bacon Wrapped Scallops might be my favorite food ever)
Mulligan's (Two locations-think Jimmy Buffett)
Bobby's (pub atmosphere, great burgers and nachos)
Driftwood (an old hotel on the beach that serves a great lunch)
Citrus Grille (good for a parents' night out, not so much for the kids)
Movie Theaters and mainstream big-box stores are a 20 minute drive. There is a very nice Publix supermarket about 10 minutes away in Sebastian with a 24 hour Walgreens next door.
Do not miss the kid's section of the Vero Beach Bookstore-it is an exceptional place. My DD7 begs to go every trip.
If you are the type that likes to stay at the resort VB is a great choice. If you are looking for more to do off-property Hilton Head is probably a better choice. That said, the actual beach in Vero is beautiful.
Last-You can always break away from Vero for a Day Trip to WDW. I can make rope drop from my in-laws house if I leave by 6:45 AM, leave after the 3:00 parade, and be home in time for a swim before dinner.
Hope this is helpful.