By choosing to travel by truck, you’re accepting the need to be able to maintain and repair that truck on the trail. What tools and spare parts you end up carrying with you are based on your experience fixing things, the needs of the vehicle you’re driving, and what its common wear items and broken parts are.
Tires are going to be the most frequently damaged part of your truck, but are also the most important part. Start with good tires which will strongly resist punctures while giving you excellent traction. At a minimum, you’ll need an air compressor, puncture repair kit, recovery tracks and an off-road jack of some kind.
After your tool kit, and all that tire stuff, you’ll need to consider basic human needs like water (one-gallon per-person, per-day, minimum), beer, shelter, and food. This is actually the easiest part of all this; a big part of the reason why 4×4 camping is so fun is that you can take so much with you compared to human-powered forms of travel. Go nuts, be comfortable, and cook nice meals.
You also need to consider fuel range. Expect your average fuel economy to halve in dirt, and plan accordingly. You either need to plan routes that include adequate gas stations, or carry extra in fuel cans. Always have more fuel with you than you think you might need.