If you have a car, there's no need to work with any tour companies. You're better off scheduling things on your own to allow for the most flexibility. The only benefit of utilizing tour companies is that most provide transportation, but that's usually from Waikiki area hotels. There are few that offer it from Ko Olina, where Aulani is located, and the ones that do are expensive.
Pearl Harbor : You can reserve USS Arizona tickets ahead and no need to pay for a tour.
Unless things have changed in the current environment, Arizona tickets need to be reserved at the 60 day mark and they go quickly. OP is already within that window if traveling in February. Because of that, you should plan to arrive at Pearl Harbor early (prior to opening) and line up in order to obtain standby tickets. Tour times are given out in order, so the further back in the line you are, the later your tour time. Afternoon tour times are challenging because its too time consuming and far to go back to Aulani, but there's not much to do in the immediate vicinity to kill time. Its also challenging to find parking there if you leave and come back. So lining up early to hopefully obtain a morning tour time is best case scenario.
Pearl Harbor is the only place that may be worthwhile to look into a tour company as they will facilitate getting your tour tickets for you. We used Discover Hawaii tours, but it included pick up and drop off from our hotel in Waikiki when we did it. I'm not sure if they have options otherwise. Our tour was for 9:15am if I recall correctly.
I'd also caution not to leave anything of value in your car as vehicle break-ins are rampant in Hawaii, and particularly so at Pearl Harbor. You're not allowed to bring bags inside, but they have a bag check outside for a couple of bucks per bag. We opted to use this rather than try to go without a bag for the day.
IMO, the North Shore and the windward side of the island are not to be missed. Kailua, which is on the windward side, had the most beautiful beaches we saw during our entire trip. Warm, clear water, powder white sand, easy swimming. The lagoon at Aulani is man-made and while its warm, there's a drop-off where it gets deep and the sand is brown and coarse and gets scorching on the feet.
You mentioned above you may be interested in the Jurassic Park stuff - that's at Kualoa Ranch, which is also on the windward side. It would be easy to plan a morning tour at the ranch and spend the afternoon in Kailua. We did this and it worked out well. I highly recommend the VIP Movie Tour at Kualoa Ranch - its a small tour in a private vehicle that takes you to the Jurassic Park stuff, including the Indominous Rex pen from Jurassic World which was very cool. I'm sure they've added and changed stuff on the tour since we were there, but we thought it was well worth it and better than the standard tour which is done a school bus with no AC. I would recommending reserving a tour time ASAP - when we were there, the tour was only offered twice a day (and we preferred the morning one) and it was limited to 12 passengers. In the current environment, I'd imagine they need to distance people so I'm not sure if its even more limited or how that's even being handled.
The North Shore was one of our favorite locations. Haleiwa is a great little town. With kids, definitely stop at Matsumoto for shave ice. Drive along the main road and you'll come upon a busy area at the beach where there are often sea turtles (Laniakea Beach). We did Waimea Valley which I would not do again. We also did the Dole Plantation which is on the way to Haleiwa. Dole is not something we'd do again either, but it was a fun touristy stop and would likely be fun for kids.
I would also suggest spending at least an evening in Waikiki for dinner. Its so different than anywhere else on the island, and there's a ton of great shopping, dining, people watching. Its easy to walk the whole strip and family friendly.
Kono's is a quick service spot with locations in both Kailua and Haleiwa. Best food we ate the whole trip! The shrimp trucks in Haleiwa are also a popular stop, though they are literally food trucks with picnic table seating. Leonard's malasadas are also a must try - they have a food truck you can see from the highway on the way to Aulani. Anna Miller's is an often recommended diner spot that's very inexpensive and known for pies near the Pearl Harbor area. We had a great buffet lunch at Kualoa Ranch, where everything was grown right at the ranch (including the meat). We enjoyed Monkeypod across the street from Aulani a few times - unique dishes and good happy hour prices. Tons of restaurants to choose from in Waikiki as well - Duke's is a classic with gorgeous views. House without a Key is recommended for something a touch nicer.
