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.