It is super hard to be on the Big Island without a car. It is a BIG island and the distances are quite far. There is no real public transportation and taxis would cost a fortune. A car would be cheaper. Akaka Falls is probably 90 minutes away from the Hilton, for instance. Kona Brewery is about half an hour or so away from the Hilton. Also, all the good beaches are a drive away. There is no beach at the Hilton.

Rent a car soon. During peak seasons they sell out and then you really will be stranded.

You are at a beautiful resort but as taitai stated if you want to see any other part of the island you will need a car. Akaka is beautiful and you will want to do it, from your hotel you would take the northern route 19, expect that it might be raining there stick those little plastic 99cent raincoats in your suitcase. We stayed in Kona for a month and every day it rained on the Hilo side of the island. We finally just went anyway and did encounter rain but it was warm.

Drive to the little town of Kona and park at the market. It is free. We bought fresh fruit there all the time. You can leave your car there and walk the town. A good place to have lunch or dinner would be Huggio's on The Rocks or Don the Beachcomer. Beautiful views at both places. A good place for breakfast is Lava Java with outside seating.
A beautiful beach is Hapuna, it is not far from your hotel, but you still need a car. It is a state park with a small consession stand. Also not far from your hotel is Lava Lava where you can eat on the beach, it is located on "A" Beach (Anachoomalu Bay) But again, you will need a car.
Note all the lava fields and when you drive up the mountain from the seashore there is a 20degree difference in temps so even if it's warm at the beach and you drive up the mountain to a coffee plantation you might want to throw in a sweater.

Yes, you should have a car.
However, while at the resort, there is a shuttle bus that will take you within the Waikoloa resort area to the Kings and Queens shops (small malls across the street from each other). There are several upscale restaurants (mostly Kings), a food court and a grocery with deli, Macaroni Grill, Starbucks (Queens). Check out the websites for both centers; there's often free entertainment (hula, concerts, etc.), and there's a weekly farmers market at the Kings shops.
http://queensmarketplace.net/en/index.php/
http://www.kingsshops.com/
The shuttle also makes a jaunt to lovely A-Bay easier (easiest is to just walk through the Marriott); the new Lava Lava restaurant is just south on the beach, or there's nice outdoor dining at the Marriott. IMO, the beach, swimming, snorkeling at Anaehoomaluare (A Bay) are much better than the Hilton "beach"; just stroll south on the beach.
There is a great petroglyph trail that you can access across the street from the Hilton (through to Kings shops; then shuttle back). Or start at Kings and take it the other direction; there are also some across the street from Kings (by the Queens parking lot), but they are not as well marked.

oops; Anaehoomalua

GEEZ! Anaehoomalu! (That's probably why it's just A-Bay)

p.s. If you have a car, the free public lot for beach access is really close to the water.

Unless your work provides you transportation to and from the hotel, reserve a car for airport pickup and to use throughout your stay.

Do you want to visit Volcanoes National Park near Hilo? (Opposite side of the island; a very long day if you don't stay overnight; do not plan to drive after dark.) You could take a helicopter tour over the island that leaves out of KOA.)

Any waterfall, especially on Big Island, is going to be dependent on the time of year.

The waterfalls on the east side are pretty much year-round:
http://www.hawaii-guide.com/big_island_of_hawaii/articles/big_island_of_hawaii_waterfalls

Yes, definitely a car is necessary on the Big Island. In the Waikoloa resort you have restaurants in the King and Queens shops such as Merriman's, Roy's, and Sansei Sushi, in addition to the restaurants that are in the Hilton. But I suggest leaving property to try local restaurants too, in Kailua-Kona, Waikoloa Village, Waimea, and if you're going to Hilo might as well eat there too. There's lots to do on the Big Island - and a lot of it involves long drives so just be prepared. If you really want to do something that is within the Hilton Waikoloa area, as Sylvia said, A-Bay is a short walk/drive away, and there is also a petroglyph trail behind one of the shopping centers (king's or queens - can't remember) which is a nice walk. Enjoy your trip!

Absolutely need a rental car on The Big Island.

You will really want to get away from the Hilton. ugh.

Your best rental car rates will be for a week, not just a day or two.

Transport to the Hilton from the airport will likely cost as much as a day's car rental.

You can see Akaka Falls and Hawaiian Volcanoes National Park in a long day from Waikoloa area, using the northern route through Waimea. Skip Volcano Winery. The wine is terrible. The coffee farms are better seen on anothe day trip taking the southern route.

Yes to a car for the whole trip. The freedom it will offer you is cost effective for quality. Sure you could just stay in the area and use the shuttle the first few days, but that is still costing you time waiting for the shuttles. With a car, you have the freedom to explore a bit while you're husband is working...say drive up to Waimea to Hawaiian Style Cafe for breakfast.

We usually reserve a car in advance, and then a week or two try to Priceline and cancel if our bid wins.

The big falls on Hawaii like Akaka and Kahuna are most certainly not seasonal. I don't know why that was a warning, and I don't know why it was suggested that especially on the big island, waterfalls are seasonal. Not true. The small falls on Maui yes. The big boys, they fall all year long.

Look for a rainbow at the bottom of Akaka Falls. Beautiful!