I've gathered a few articles together to help you learn what you need to know to determine the right baseline pressures to experiment.
Expedition Portal has just published a great tech article on airing down your tires for off-road use, with some tire footprint measurements and analysis.
Rockcrawler shares their thoughts in this article.
Warn shares their thoughts on their blog.
4x4NOW put together a great article with illustrations to help visualize some points I've struggled to verbally explain around the campfire:
Overland Journal has a good video on YouTube if you're interested in seeing the above-mentioned concepts in motion out on the trail:
There are a number of options available from the aftermarket to make airing down your tires easy and convenient. After trying several different options, I found that I preferred the Staun adjustable deflators over all others and I've had great luck with them for many years now.
A cheaper alternative, and one which I use when I need to fine-tune my tires to an even lower pressure for extremely difficult terrain, is a set of these deflators. They don't automatically shut off at a preset pressure like the Stauns do, but they allow you to quickly and easily check the tires' pressure with a pressure gauge as they're deflating, without needing to unscrew them - just hold your pressure gauge up to them and repeatedly check your pressure until it's where you want it.
If you would prefer to manually air down each tire one at a time, the best deflator and gauge combo is this one from ARB.
Of course, once you've aired down and had fun on the trails, you'll eventually need to get back home; if that involves driving on road, you must air your tires back up. There are countless compressors out there to choose from, and this is one of those products where price and performance vary widely. Only you can decide how much speed you can afford, but I think the "sweet spot" for the price/performance ratio is with the Smittybilt 2781 MV-90 SuperFlow Compressor. It's well built, compact, reasonably fast, and surprisingly affordable compared to its direct competitors.
VIAIR, ARB, and numerous other companies make a wide variety of 4x4 air compressors so feel free to shop around for a unit that suits your needs the best. All 4x4 Internet forums have threads you can search for in which compressors are reviewed and compared. ARB produced this short video showing their air up/air down products in real-world use:
Don't be afraid to experiment with different inflation levels - just be mindful of how off-camber and turning at speed can try to peel the tire's bead off the rim and cause a sudden complete deflation of the tire. While airing down to 10psi might be a perfectly good starting point for a lightweight vehicle like a Suzuki Samurai with large tires, a heavier vehicle like a Wrangler Unlimited will be better off at 20psi.
Pay attention to your tires, how their sidewalls bulge, how the tread wraps around rocks, and how the steering feels; adjust your air pressure as necessary. In particularly loose, soft terrain such a snow, you can safely air down to single-digit pressures - just be particularly attentive to keep the tires pointed as straight as possible to avoid blowing the bead. This is a particularly effective trick to extracting your stuck vehicle from snow or sand without breaking out the jack, shovel, or winch.