Fishing is very much about respecting the fish. This means that if you are going to keep the fish you must kill it as quickly as possible. If you intend to take fish home then I thoroughly recommend carrying a heavy stick or 'priest' (so called because it delivers 'last rites' – no kidding). Knock the fish smartly on the top of the head, above the eyes. All we are doing here is knocking the fish unconscious. The fish will die shortly afterwards through oxygen starvation – they can only breathe in water.
It is very important that you understand how to correctly handle fish. Firstly you must wet your hands. This is partly to wet them and partly to reduce heat shock. Fish have a protective slime. Dry hands will remove this and make the fish susceptible to disease.
Handle the fish away from the eyes and gills, gently but firmly. If you hold the fish upside-down it will struggle less, which is rather interesting. I can't tell you why: only that it is so. I have often wondered why it is. I think I would struggle like hell, but there; I am not a fish, no doubt if I were, I too would lie in a calm trance-like state.
Another thing to be aware of is spines and teeth. Trout don't have them; well actually they have teeth, but are extremely unlikely to use them to bite you. Some fish may do so. I have yet to be attacked by a landed fish, and am looking forward to the day it happens, albeit with some trepidation. Some fish are spiny and being spiked is not very pleasant, by all accounts, and being spiked by a poisonous spine, is downright unpleasant.
A fish to be returned should be unhooked as quickly as possible. Carry some surgical forceps for the task. I certainly hope that you are fishing barbless – you will catch more. If the fish is hooked deeply it is best to cut the leader as near to the fly as possible and return the fish. Messing around will only cause the fish to bleed. A bleeding fish is unlikely to survive. The hook will corrode very quickly, often in a matter of days, especially for either deeply hooked fish or fish in saltwater.
In order to return the fish gently place him in the water, cradling him in your hands. Take your time. They deserve it. Allow the fish time to recover. If there is a current place the fish pointing head upstream. It is necessary for water to flow over the gills in order for the fish to breathe. In stillwaters you may have to walk the fish forwards in order to help it breathe. I get the impression that as soon as the fish realises that you are going to return it, it takes it's time and trusts you. It certainly appears to be the case.
Being capable of returning fish unharmed is essential.