I've heard good things about Hike Kauai with Me over on TripAdvisor but have no personal experience since developing mobility problems two years ago which cut my hiking down to zilch. Might want to check them out.
You'll need to take a taxi or shuttle from the airport, since you cannot take luggage on the Kauai Bus (limited schedule).
Both the Kauai Marriott in Lihue and I think Kauai Beach Resort have a free shuttle. The Marriott has several restaurants onsite and there are two small shopping centers with restaurants in walking distance.

Jamie99, thank you. Would you recommend one hotel over the hotel in terms of locations? I'm interested in being able to walk to places.
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Hey there, I'm a Oahu born and bred girl and I'm going to be very blunt with you. Please know that Waikiki will not give you an authentic view of the islands as it's completely culturally divorced from the rest of the island. It's also very expensive and the traffic is rough. I highly reccomend staying in a condo somewhere between Ala Moana shopping mall and the downtown area...china town, kakaako, ward, and most of Ala Moana Blvd is a good bet. Staying just outside of Waikiki will give you access to better restaurants (China town is definitely home to Oahus foodie scene) and there is ample public transportation. I would suggest a combination of the Bus and Uber to get most places. Avoiding Waikiki will likely shave off 15 to 20 minutes off of your transportation time for most of the must see sights. You can do so and see so much without a car on Oahu.
Some suggested day plans.
Day 1: Breakfast in Chinatown (scratch kitchen for contemporary American with local Hawaii flavors, or go for dim sum at Legends)
explore downtown and Chinatown area on foot (there are some pretty cool historical buildings and some awesome galleries) and check out Iolani palace
Take the bus or use Uber to head to lush Manoa Valley --- if you are hungry or need a coffee boost stop at Morning Glass (excellent Kona and Kau coffee and great sandwiches) then spend your afternoon at Lyon Botanical Gardens (be sure to skip Manoa falls since they will be a trickle during the summer lol) be sure to take all the little trails --- its an absolutely amazing collection of plants and the mountains that surround it are so gorgeous! Uber or bus back into town.
Grab dinner downtown- for contemporary riffs on local favorites in the downtown area I suggest Pig and the Lady, Senia, Fete, Lucky Belly or Livestock Tavern. For excellent Japanese -sushi sasabune for sushi or nanzan Giro Giro for omakase.

Day 2:head to the windward side via BUS in the morning for Hiking and breakfast---do them in whatever order works best for you --- though I'd suggest trying to get on the trail before 10:30am or wear lots of sunscreen. Highly reccomend Mokes for breakfast for their lilikoi pancakes :)
For trails I'd suggest Lanikai Pillboxes (depending on what you are used to, I would grade this trail somewhere between easy and moderate---there are a couple of eroded steeper sections, but nothing too bad as long as you have appropriate footwear). Be sure to go to at least the 2nd pill box to have more of the trail to yourself---technically the trail continues all the way down to enchanted lakes if you are so inclined. The trail offers views of Kailua Bay, Waimanalo, the Mokolua islands, the Koolaus, and Mt. Olomana.
after you finish your hike wander about a quarter mile down the road for Lanikai Beach Access. Find a shady spot if you can and enjoy the cool water and gorgeous views of the Mokes (mokuluas)
For a great and local lunch grab a plate lunch at K and K Barbeque (Korean style bbq basically) and get a Teri beef plate or a BBQ chicken plate....or if you want amazing fried chicken go to Fatboys in Kailua town and grab a Garlic Chicken Plate. If you want to have something a little healthier Lanikai Juice (also in Kailua town center) has great fresh fruit bowls and smoothies.
if you decide to make a full day of it in Kailua---there are lots of cute little shops and boutiques and several pretty good restaurants for dinner ( cactus, fromagios,lemongrass, and buzzes are probably the best restaurants for dinner)

Day 3:
Take the bus that goes to the University of Hawaii . Get off the bus near Pucks alley (just ask the bus driver for help) and then grab an amazing breakfast at Nook--- pork belly eggs Benedict yes please :)
Then grab an Uber and head up towards tantalus(Hawaii's only historic road) on the way up ask your driver to go via Round top drive and make a quick stop at the lookout at Puu Ualakaa State Park and maybe pop into Nutridge (historical house) for photos then continue via uber another 4 miles or so to the top of the mountain. Ask your uber to drop you off just before the one lane bridge where the Kalawahine trail begins. From here you can take a series of trails that snake down along the mountain side offering gorgeous views of Pauoa and Manoa Valley. You will also pass through a very special area of endemic native plants that is maintained by the university ---looks a little like jurassic park. Take the Kalawahine trail to Manoa Cliff trail then cross the road and take the Moleka trail to the Makiki Valley Loop trail ending at the nature center. Total distance is probably 7 miles or so, but most will be relatively flat or downhill, there will only be a few areas with uphill sections. The views will be quite beautiful for most of the hike. Walk down the road about a quarter mile to Nehoa street to catch Bus back into town
Day 5:
take bus 52 to Haleiwa/wahiwa (this one is a longer bus ride) and head up to the North shore---about half the drive is scenic, half is less so lol. Enjoy walking the town of Haleiwa and walk Ali'i beach (look for turtles resting near the tides---i see turtles 9 out of 10 times I walk that beach) grab Breakfast at Cafe Haleiwa....if you are adventurous go on one of the shark cage dive tours (needs to be booked in advance)...ive done several shark dives and would reccomend the shark dive in Haleiwa to anyone. You are in a sturdy cage and the sharks are relatively small and not aggressive---its just a beautiful sort of surreal experience --- this is not the terror inducing great white shark experience like you are on shark week (done those also---wouldnt reccomend those generally lol). From Haleiwa you can catch the bus (#55) and gradually make your way down the coast line--- i believe the bus comes every 30 min or so---but depending on your interests you might consider getting off at Waimea beach (for a little beach time) or Laie (for polynesian cultural center) Take the bus into Kaneohe and get off at the Haiku bus stop and walk to Haleiwa Joes for cocktails and a meal with a view :) bus or uber (approx $25) back to Honolulu...

hopefully this gives you a little taste of the possibilities beyond Waikiki :) feel free to message me if you want any other ideas

Bookmarked...great detailed post!

With no car, I would pick the Marriott, stayed there quite a few times, Kauai Beach Resort is kind of set off by itself. I've visited Hanalei Bay Resort a few times to see friends, it is in Princeville and is a beautiful place, but only a few restaurants in walking distance. Most restaurants are in Hanalei which is not walking distance so you would need to take taxis.

Thank you so much, trailandtide; appreciate your time with all the details. And thanks to you too, Jamie99.

bookmarking as well.