DISCLAIMER:
I do not know what I am doing. I have never had any instruction in tree-climbing techniques, rope-work, or rock-climbing. It's all from Google, and meeting people out there and asking questions. I've not fallen out of a tree yet, but I'm bound to eventually, and my good fortune so far should not be seen as grounds for relying on my advice to any extent...
UPDATE: Since writing this, I've made a heap of videos on the subject.
Ropes
Ropes are basically static or dynamic. A static one does not, really, stretch any if it catches you when you fall. Dynamic ones do. This means you don't get a nasty injury from the sudden stop with a dynamic rope, because it gives nicely. On the other hand, it makes the whole set-up a bit spongey and possibly less energy efficient.
Harness
Essentially there are climbing harnesses and arborist harnesses. I bought the cheapest climbing harness I could find, because I didn't anticipate getting hooked the way I have. Arborist harnesses are really specialist bits of professional gear, cost lots, and are only available from professional suppliers really. The interesting difference though, and it's an important one, is that (rock) climbing harnesses have a single point of attachment at the front-centre (that is, the belay loop and the two parts of the harness - together - that the belay loop connects to.) Arborist harnesses have two big happy load-bearing metal rings on your hips, and a bridge between them, where you can attach. This means you can hang from your tie-off point and lean left and right, because the fixing-point moves along the bridge, and you can use a lanyard to clip to one hip, round the bole (trunk) or limb, and to the other ring, to secure yourself comfortably and stably while re-pitching.
How to climb
Essentially, you are climbing a rope, not a tree - although you use the tree to help you get up the rope. There are basically two categories of rope-climbing systems: SRT and DRT. SRT is Single Rope Technique and means any system where you tie the rope to something, eg your mate ties it to a high limb, or you throw it over a limb and tie it off to something solid on the ground (or the trunk, ensuring it won't slip upwards). Then you climb up the rope by whatever of various means and with whichever skills and bit of kit you might possess. Things to look up might be foot-locking, ascenders, friction hitch, and so on.
DRT is Double Rope Technique. It still only uses one rope (!) but this time you throw the rope over a limb, attach it to yourself, and then pull yourself up with the trailing end. Essentially you are climbing the trailing side of the rope, but as you move up, so that side comes down. This means you have created a 2:1 pulley system over the limb, so it takes half the energy to pull, but you only advance half as far. That's a big advantage though. Unless you are a gibbon. The other advantage, and it's significant, is that when you get up high in the tree, you can untie the rope, throw it over a higher limb, tie-off again and keep climbing. (Though, obviously, have a system so that you cannot drop the rope on the ground (stranding you), and so that you cannot fall out of the tree (breaking you). This latter is easily achieved with a lanyard made from a climbing sling (ok but usually not dynamic) or a short (about 3m) length of rope. If you make one of these (illustrated), you can easily adjust the length of it so you are secure. the red cord is a "prusik knot". It is made from 5mm climbing cord joined to a karabiner with a Double Fisherman's knot. The pulley isn’t necessary but REALLY helps as it means you can tighten the lanyard with one hand. The end of the rope has a Stopper Knot for safety, and the left-hand karabiner is attached with a similar knot that which cinches to the krab under load. Google that lot too.
Putting the rope higher and carrying on is called re-pitching. It is now a multi-pitch climb. Each leg of the climb being one pitch.
How I climb DRT
I began climbing on a set-up of Blake's Hitches that self advance. Look here for how that works! It's effective and cheap, and, like the hitch-climber pulley system below, you can descend on it as well as climb. However, look at it! It's a bit of a faff if you want to re-pitch. Also, I found it was quite easy to tie a Blake's Hitch wrongly. It's not as lethal as that sounds because it's very obvious when it's wrong, but it's a massive pain.
These days I climb with a Hitch Climber Pulley system, as shown very clearly here on Climbing Arborist. However, I am a flimsy old man and find it very hard to pull myself up, so I also use a Petzl Foot Ascender to pull the rope down (so pulling me up). (A cheaper, but way less cool, alternative would be to make a long-ish prusik loop, attached to the "down" rope, and stand in that.)
You can use one of many flavours of friction hitch - Distel Hitch, Blake's Hitch, etc. I use a Valdotaine Tresse, because it is very sensitive, seems to descend easily and rarely locks up.
Amusingly, if you have a hand-ascender, you can attach it to the down rope above the friction hitch, pass the trailing end of the rope through it (or, more conveniently, through a krab attached to it). Then, if you pull down on the down rope, you've made a THREE to one pulley system. For weaklings.
The basic multi-pitch DRT climbing system is clearly explained here on Climbing Arborist, although the video and sound quality isn't great.
Getting down again
If you are using any system with a friction hitch, you can descend without changing the setup at all. Just remember that where ropes rub together, where a lot of rope rubs against a single, small piece of rope the single, small piece of rope will quickly heat up and potentially fail. So descend slowly and monitor the friction hitch. For a small investment of time, you could dismantle the climbing set up, and abseil down with a nice metal belay device which, again, will heat up - but the only danger is burning yourself as it will not burn itself. A very good idea is to use a prusik as a backup on the belay device. Watch Get Out On A Rock to see what I mean.
Limbwalking
So you're coming down on your friction-hitch setup, but hang on!
Now you've got a nice high anchor in the tree, and you can use your easily-adjusted hitch-climber pulley setup to walk out along limbs. You can get anywhere! How cool is that?! This is where the Valdotaine Tresse really comes into its own.
That's probably about it! Have a nice time!
Really good resources:
Tree Climbing Field Guide
Cornell Tree Climbing YouTube guide
Climbing Arborist YouTube guide
Arborpod YouTube channel
Animated Knots (they also have an app, which is brilliant in trees)