Need advice as to which island to visit. Our interests are hiking, eating, learning history and culture, walking tours or other tours (as long as they are cheap), and seeing interesting or strange landscapes and buildings.

I'd like to walk through the beaches but not sit at one for an extended time, maybe just an hour at a time. Certainly I'm interested in seeing beautiful beaches obviously but my idea of fun is not to just sit at them for hours. Not interested in swimming in any form. Wading around in a relaxed place for a few minutes, yes, but no actual swimming, snorkeling, surfing, shark visits, etc.

As far as hiking goes, we don't own cars so we can walk a few miles on any day but we are not adventure hikers that can do really strenuous long hikes, nor will we have brought the gear to do such things.

As for eating, I am fine with anything but my gf has issues with gluten, eggs, dairy. I personally want to try all the most legit stuff out there. That doesn't mean expensive.

We will rent a car at some point but would rather be able to not rent for some of the days just to save money. Where can we go on just public transportation?

Coming from the mainland of course, New York. I'm wondering is Oahu the cheapest/only airport to fly into from the mainland?

The only island on which you can rely on public transportation to get you around is Oahu, which also satisfies all your other "wants" - eating, history, culture and beaches. I assume there is hiking, although I have never hiked on that island. Oahu is a little bit of everything - city, Waikiki (developed beach- dense, many hotels and condos and shops and retaurants, but also the beach), Pearl Harbor, gorgeous beaches (Kailua, Lanikai, Sunset, Waimea). Just the drive from Kailua on the east to Haleiwa on the north shore is spectacular. There are other "cultural" destinations, although I've never "done" any of them, despite multiple trips.

Oahu is the island that best fits most of your requests.

Staying any place other than Waikiki/Honolulu (on the island of Oahu) on any other island requires a rental car full time (maybe you could make Maui work staying in Lahaina but not nearly as much going on there).

For culture and history I highly recommend doing the Iolani Palace tour. There is also Bishop Museum and Foster Botanical Gardens. All accessibly on The Bus (the public transportation system).

I'm wondering is Oahu the cheapest/only airport to fly into from the mainland?

No, there are west coast flights directly to other islands coming out of Seattle (and I assume other West Coast cities)? Years ago you had to go thru Honolulu, but no longer true.

Yes, we can fly nonstop from the Bay Area to all the main islands - Maui, Oahu, Kauai, Big Island.

Last month DW and I flew from PDX to Maui (OGG). It was our first trip to Hawaii (my 50th state). We liked Maui well enough but if we go back to Hawaii again, we'll probably go to Oahu to see Honolulu and Pearl Harbor. There are over one million people on Oahu but less than 200K on Maui.
There are 7 day cruises that sail out of Honolulu that visit several islands including Maui. If you want to see several islands, a cruise would be better than several inter island flights IMO.
I have no desire to see the active volcano on the island of Hawaii.

Because of the issues with transport (need public transportation) and your wish to learn about the history and culture, I agree that Oahu is the island for you.

I disagree that a cruise would be the best way to see several islands. On a cruise, you only have a matter of hours on each island. The islands are all large enough that you wouldn't be able to see much in the few hours you are there.

We would need to know how long they have for this trip total (I don't think they mentioned that?) to say if only one island, or more than one island could work in their time frame.

Oahu is the only island with decent public transportation, the other islands, the buses where they exist travel to places the locals want/need to visit, not the tourists.
Even on Oahu you probably want to rent a car for a day or two to drive around the island (bus takes forever), etc.

I'd be staying for 2 weeks.

Also, I just don't want to rent a car for the entire trip basically to save money, as I originally say: "not rent for some of the days just to save money." So I'd be fine on any of the islands that do require a car, but I want to know which ones have the best sights to see based on my taste. But what I hear from everyone is that I can get away with public transportation for a few days in Oahu.

I want to know which ones have the best sights to see based on my taste

And the answer's Oahu not only because of transportation but because of this:

learning history and culture, walking tours or other tours, and seeing interesting or strange landscapes and buildings.

That request just fits Oahu more closely than the other islands. I also think Oahu can be the most reasonably priced.

With two weeks you could visit two islands, Oahu for the culture and dining and public transport and one other. When is your trip? My personal favorite is Kauai, but some of the North Shore is closed (past Hanalei) due to recent major flooding, but it has good hiking on other parts of the island, such as Waimea Canyon and Koke'e State Park, SLeeping Giant and others.

We're thinking January. If not, then April.

I have been to the other main islands several times (Maui, the Big Island, Kauai) and I agree Oahu would be the best for cultural (Bishop Museum, Arizona Memorial, Iolani Palace, Chinatown, Byodo-in Temple, etc). There are interesting natural sites such as Diamond Head, Hanauma Bay, the beauty of the North Shore, etc. It is also in my opinion one of the prettiest and most tropical (along with Kauai).

Maui is more geared to large resorts with access to the beaches and water activities such as snorkeling, diving, beach going, etc. The Big Island is ... well ... very big and you would need a car and it is difficult because to get around to various places requires sometimes long drives.

Besides Oahu, what other island should I go to since I have 2 weeks?

Also, it doesn't matter if I need a car or not, seriously please ignore that. Everyone seems to think I don't want a car at all when I've said twice that I am only going to not rent a car for a few days to save money. Yes, for the third time I am going to rent a car, just not for the entire time. "Where can we go on just public transportation" does not mean I am going to rely on public transportation for 2 weeks, it means "where in Hawaii can I go that I can get away with that for a few days?" The answer is Oahu, thanks. Yes I am even going to rent a car in Oahu, I'm just saving a few days worth of money.

Personally I would go to Kauai as a second choice. You mentioned cultural things to do. Maybe check with some of the bigger venues that cater to luau. The main city is Lihue. They may have some local luau activities as well. Kauai is not as developed and the main attractions are the beautiful and stunning landscapes. I recommend doing a helicopter tour. The is no public transport to speak of so it is good that you will be renting a car.

I personally think the "culture" on Maui might be too focused on the ocean activities as there is really not too much other than a cowboy town called Makawao. You might find the Haleaakala Volcano interesting in that it is like a lunar landing site!

The Big Island requires hours of driving because the various tourist sights are long distances away from each other. The side of the island with the Kileawea volcano eruption will limit your ability to see that side of the island right now.

Kauai is my favorite, and I would recommend it highly. The National Tropical Botanical Garden is there and there are a number of tours you can take of the stunning Allerton Gardens.

I consider some of things available in Lahaina on Maui to be plenty "cultural". The monarchy was based there all not that long ago in history.
Lahaina Restoration Foundation, Preserving Maui's History

There's a self-guided walking tour of historic buildings in the central part of Lahaina town.

The show called U'lalena at Maui Theater is very interesting and entertaining: https://www.mauitheatre.com/

Of course Old Lahaina Luau or Feast of Lele for a more tourist-oriented but fun night of island music and dance and food.

You can find bits of history on all of the islands. Oahu (IMO) has the most, but Suze is right there is history at Lahina on Maui. The last home of the last queen of Hawaii is located in what is now the Allerton Gardens on Kauai.

January is winter in Hawaii, too. Not a prime time to visit the northern shores of Kauai or Oahu -- rain, wind, high surf. You would probably want to stay south on either of those two islands and tour north. Winter is also Whale Season, best observed from Maui or Big Island of Hawaii -- and probably already heavily booked on Maui. April is usually nice all over; May, nicer still.

Two weeks is enough to enjoy two islands. You can split time however you want. It sounds like you will fly via HNL and that you're interested in staying Honolulu. You might make a list of what interests you on a second island; think how long you'll need to see/do that; spread your time accordingly.

Day One in Hawaii is somewhat lost as you adjust to the time change. You'll wake before dawn and fade early. Each transit between islands will consume 4 - 5 hours, door to door -- highways, airport time, rental car, finding your way, etc.

Booking a condo for some of your stay is a good way to save. You can 'eat in' some of the time. Few condos charge the daily 'resort fees' and parking fees that hotels demand.

I'd skip a car within Honolulu, where traffic is bad. You can walk, use the public bus system, even hail a cab for a small fare over the short distances. Rent a car for at least one day to drive Oahu's coast. I would want a car full time on any of the other islands. Small is fine; just 'wheels'.

If I'm not mistaken, you can get to the starting point for Manoa Falls via bus. It should be a short hike up to the falls. Just remember to bring some water and a mosquito repellant.