Food: I'm one of those crazy people who packs meat when we go on vacations where we'll have access to a kitchen. Typically, we bring bacon, sausage, hamburger, steaks, and chicken and pack it all in our 8-can soft-sided cooler, along with some blue ice packs. (Don't pack ice in a baggie if you plan to carry-on, as ice turns into a liquid and is banned via TSA.) HRA has no grills or ovens, of course, but I just planned meals around it.
If bringing meat along isn't your cup of tea, just throwing a few dry items into your luggage can still save money. For example, bagged chips (e.g Lays, Doritos) are really expensive in the Caribbean (e.g. $8 for a medium bag), so bring snack items if you can fit them. If chips are too bulky, Pringles, Combos and crackers are great alternatives.
Other things you might bring: coffee & filters, condiment, seasoning, sugar, and jelly packets, chili mixes, cans or bags of tuna, summer sausage, boxed pasta and potato side dishes, peanut butter, TidePods, dryer sheets, boxed cookies, candy, whole onions, huggies (for drinks), a wad of sandwich and gallon sized baggies for keeping leftovers and packing picnic lunches.
Since you're paying for a taxi at HRA, come armed with a grocery list and divide and conquer with your spouse/kids. Even if you plan to eat out every lunch and dinner, just buying breakfast items, fruit, snacks, desserts, and drinks will still be worthwhile. Hint: Imported beer is very expensive, but the local beer (Kalik) is fine, according to my husband.
FYI, we packed lunches and drinks (including water, beer and mixed drinks) in our cooler and brought it to the beach/onsite pool/water park nearly every day. No one even batted an eye at us. (We ate at our chairs, not at a table at the restaurant). When we did eat at one of the outside waterpark restaurants, we found the food to be good, and prices to be comparable to any other tourist trap. It's the VAT that pushes the price way up.
We also ate at Margaritaville (fine), Anthony's (good; short walk away), and Curly's (delish and low priced) at the fish fry in Nassau
Transportation: We used Sandra's and arranged ahead of time to pick up groceries on the way. It worked out really well. You really don't have to pre-arrange this, as taxis are plentiful everywhere, but it was nice knowing what we'd be paying ahead of time for the trip + grocery. Wouldn't pre-book taxis on the way back because we were ready to go 30 minutes before our time. We'd have rather grabbed a taxi and headed out immediately than waiting around for our designated time.
To get to/from the water park, look for shuttles that have a turquoise band, as those go to HRA. Other shuttles don't stop at HRA, but you'll figure it out.
To get to/from Nassau, either take a taxi or use the water taxi that is a very short walk away.
Beach chairs: Not necessary. The water parks have plenty of free chairs, and plenty of shade. Note that we traveled in low season. If you really need shade, and you can't find them, consider renting one of the many cabanas. If you go to the beach, go to Cove Beach. You can rent a beach chair, if you want. If you go to Atlantis Beach, you can rent a cheap chair from one of the vendors, but you'll be bothered by beach vendors all day.
Wifi: Had terrible service with Pocket Wifi, as it quit working after day 1. Couldn't get our money back because we'd pre-paid it outside the 3 month window where you can dispute charges. We ended up paying for the onsite wifi, which was expensive, but worked. We had no trouble connecting at the pool.
Gym: Onsite gym was adequate. There's a much nicer, humongous gym somewhere on site, but you have to pay extra for it.
Activities we enjoyed: Snorkeling the Dig (their huge aquarium), but it was pretty expensive. Also, had an in-room massage by Antoinette at Wellness Spa. Reasonable prices. The water park is the main attraction, though.
Hint: Lockers are very expensive at the water park, so if you plan to enjoy the water, bring an AquaPak to store your room key/cash and leave your other valuables at home.