I'm putting these in order of what I think is greatest to least important. When in doubt about any of this, buy from a reputable off-road specific vendor like the vendors that support this forum, or
Off-Road Warehouse.
Fire Extinguisher:
The biggest that you can fit and make secure in an easy to access location. One on each side of the vehicle is uncommon and a good idea.
First Aid kits:
One for humans, one for animals if along with, and one for the vehicle. A first aid kit for a vehicle will be extra fluids, fuses, possibly some "mission critical bolts", maybe some sealants and hose clamps, etc.
Tires:
A quality tire pressure gauge - Something like this:
https://www.pegasusautoracing.com/productdetails.asp?RecID=25352 In gauges you tend to get exactly what you pay for. A $15 tire chuck with a gauge from Harbor Freight is badly inferior to a gauge like above. I've bought those HF tire chucks before, and I always replace the gauge in them with one from McMaster.
Viair's offering is likely better than the HF part, but not as good as the Longacre part. Buy the gauge with a range that puts your normal street pressure at between 1/2 and 3/4's of that range. So a 0-60psi range is about ideal for a roughly 30psi street pressure. Most gauges are the most accurate in the middle of their range.
Some kind of 12 volt electric compressor - I'll suggest one of the MV-50/1050 family as the lowest cost option with Viair & similar compressors being better but more expensive options. If it plugs into the lighter socket and doesn't have clips to connect directly to the vehicle battery you don't want it.
Automatic air-down tools are nice, but not a necessity. Those hose manifolds are painful and bulky - BUT if you have a locker in the rear axle it is advisable to have a hose to equalize the tire pressures on that axle when you're preparing to go back onto pavement. I used one of the self-coiling air hoses from Amflo with two self-retaining tire chucks for this.
Tire plugging kit
Food, water, & Clothing:
Stock the vehicle with extra of all of this. On long trips one of my defaults is several sets of clean, dry socks in doubled zip-lock bags. In modern terms it would be called a "bug-out bag" I guess, but I call it my "Boy Scout Bag" (i.e. "always be prepared"). The socks, first aid kit, compass, extra hats, clothing and gloves + too much else to list here all live in this day pack sized bag that stays in the vehicle. Search "Ten Essentials" for some input on what else should be in that bag or stash. Make particular note of the Eleventh Essential that some mention and others don't, TP. I keep mine in a zip-lock bag to keep it clean and still rolled up.
Radio:
see BadSquatch's suggestion above, that's just the tip of that rabbit hole. there are threads here to go into more detail. Don't worry about it being confusing, it sorts itself out with a little time and use, and there are plenty of knowledgeable folks who can help in the mean time.
Recovery Strap:
Note that there are two very different kinds of straps sold. One is a tow strap and these are not designed to stretch. The other is a recovery or "snatch" strap and they are designed to stretch. You want this latter type. Under no circumstances should you buy a strap that has metal hooks on it. Ever.
Shovel:
full size if possible, 'D' handle is a decent compromise, folding is a better than nothing option. If forced to go with the last, strongly suggest buying the shovel made by Glock. It is the best folding shovel that I've ever used.
It used to be that a Hi-Lift Jack always made these lists. In the hands of the unskilled or unknowledgeable in their use they are dangerous! In the hands of those skilled and knowing in their use they're still dangerous!
I'm sure others can add to these lists, but I should go back to work now.
