Well, it looks like a geography lesson is in order. The following islands are smaller and have fewer people than Kauai (area - 550 sq.mi.; population 67,000) so they will meet your desire to avoid crowds and large cities:

Molokai - area - 260 sq.mi.; population 7,300

Lanai - area 140 sq.mi.; population 3,200

Niihau - area 70 sq. mi.; population 200

Kahoolawe - area 45 sq. mi.; population 0 (zero)

The other more well know islands of Oahu, Maui and Hawaii are all significantly larger with dense population so I doubt you'll find the solitude you seeking on any of those.

Investigate spending the entire time on the island of Hawaii aka: The Big Island (TBI)

And cross Nihau and Kahoolawe off your list.

I can't see more than a couple of nights on Lanai. Unless you are a fanatic (rich) golfer.

I agree you should look at the Big Island and I couldn't spend more than a couple of nights on Lanai either. The only one I haven't visited is Molokai.

Another vote for Big Island of Hawaii. You'll want to stay both Kohala area (NW) and near Volcanoes National Park (SE). Ten to 14 nights should be just fine. Be prepared: This is NOT a 'typically tropical' island like Kauai, Oahu and Maui, and it doesn't have many easy-access sandy beaches. It does have the best sealife of all the islands and more mini-climates than you'll be able to investigate in a short stay.

What time of year will you visit Hawaii?

"No big cities or too crowded places..."
The only "big" city in the state of Hawaii is Honolulu.

"...we like wild nature, hiking, beaches, and different activities."
I would suggest Maui. It has everything you're apparently looking for. Some think it is crowded or too commercialized, but that is a matter of opinion.

There is a certainty on these forums. Everyone has their own opinion. I would love spending a week on Lanai. I would be equally satisfied spending 4 nights exploring a part of Hawaii. Give me 5 nights in Makena and 5 nights in Hana? It's all good.

We spent 10 nights on the island of Hawaii, and it was wonderful. We spent 3 nights near Kona, 2 nights near the volcano, and 5 nights on the Kohala Coast. We felt like this was the perfect combination, and we also enjoy the things you listed.

Kauai and the Big Island are our favorite Hawaiian islands.

Have fun planning!

My vote goes for the big island as well. We splurged on a hotel for 2 nights on Kona coast, then went on to spend 3 nights in a magical B and B cottage in Volcano (Nihea cottage, Volcano Places)--very remote at the edge of a state rain forest and finished with 5 nights on the Kohala coast (Mauna Kea). We also thought this was the perfect combination and wish we had had a bit more time.

I suggest the Big Island as well, which is my favorite island in the world! Make Hilo one of your bases, if you have a car, as you can explore the Volcanoes National Park as a day-trip from Hilo, as well as lots of other beautiful spots.