I am amazed at how few trail riders I meet carry a spare tire or a repair kit. If you drive with any regularity on a mountain slope or a river bottom, a sharp rock or beaver stick will find a way into your tire. A spare is great if you have the room, but most situations can be remedied with a tire repair kit and a portable 12-volt air compressor.
The compressor will get air in the tire to find the leak and fill it once the repair is made to get you back on the trail. Portable air is also your only hope if you hit a rock and blow a bead.
Start with a tire breaker bar and the right size lug socket to loosen the nuts on the wheel, and carry a small hydraulic jack. The jack is extra handy when you don't have three guys to lift the ATV.
If you've never used a tire plug kit, they take a little practice. Make sure you have extra plugs on hand once you go through the four or five that come with most kits.
Although cans of compressed sealant may be tempting to try, the stuff can make a permanent repair much more difficult and expensive. A tire repair shop once charged me an extra $45 because I used the goo.
One of the best kits I've found is the Slime 70004 Power Spare 48 Piece Tire Repair kit. It includes 30 repair plugs, a compressor and anything else you need to get a tire to hold air on the rim.
You can get one on Amazon for $60. The kit will also work well for your truck and trailer. Most repairs can be made without the Slime goo; use it as a last resort.