Definitely get off property, the island is lovely with lots of other sites to see. Conch Shack (and next door at Horse eyed jacks) are great places to have lunch by the beach and get a different view of the island. You can also head over to the Grace Bay area before or after and check out the Regent Village/Salt Mills shops/Ports of Call shops. You can park in one spot and do the Salt Mills and Regent Village and then drive 1/2 mile down to the Ports of Call area. We have some amazing artists on island in many different fields, but I don't think any of them sell their work at Beaches.
My favorite places for unique jewelry/art/crafts: Anna's Gallery at the Salt Mills and Brenda's Ceramic studio a few doors down (I think it's Paradise gifts, I just know it as Brenda's shop, but it has beautiful ceramic ware and jewelry in the window) Anna Bourne has her amazing vivid silk art prints and great jewelry that is unique and fun different gifts, Brenda is a phenomenal ceramic artist, and has big and then small pieces that are ceramic shells and sanddollars in tiles that can be easily taken home and one of a kind. In the Regent Village, check out FOTTAC (flavors of the Turks and Caicos) with our local Bambarra Rum, as well as handmade baskets, bath salts from Middle Caicos and Sea Salt from Middle Caicos and Salt Cay! There are a few other galleries/cool gift shops in the Regent Village as well. If someone is looking for Cuban Cigars, then head to the Goldsmith in the Regent Village (my friend says her husband finds the best deals and quality there, although it looks like we may have a cigar shop opening soon in the Salt Mills). We also have some new coffee shops etc opening in December so that should be fun. Just down from Regent Village is the Pizza Pizza Plaza, and Island Scoop which has great homemade ice cream in very fun flavors (love the mojito ice cream). They also have a little gift shop next door with hand made/fair trade and organic items.
Ports of Call is down the road with more gift shops. Dive Provo always has some really unique items in their shop and around the corner from Ports of Call is the Patty Place with fabulous Jamaican style patties for a quick snack or two...Yum! A Little further is Caicos Plaza, with the French Caicos Bakery and Rumours another lovely little shop with unique items.
If you are going to go out, shop in the morning (most shops open after 10) or right before heading home and then head over to the conch shack. If you still want to explore more of the island, Chalk Sound is stunningly beautiful, I recommend going to Las Brisas on the water. They make great mudslides/mojitos and have a pool to swim in or paddle boats to go out on Chalk Sound. Some guides will tell to to go to "the Hole" don't waste your vacation time....it's just a hole on some land with water in it LOL...kinda cool but not worth a separate trip.
The island is very safe (although like anywhere in the world, don't leave valuables in a locked car etc), and really easy to get around (we have 3 main paved roads...and no stop lights) Do check it out and see what's outside of Beaches! We are a US dollar economy, so you don't have to worry about currency exchange! And don't forget your cameras!
Hope that gives you some ideas. Grace Bay Car Rentals will drop the car off at the resort and have always been highly recommended on Trip Advisor. We also have an Avis franchise here as well.