There are four basic ways to approach this, lead taming, para taming, peace taming, and honour taming. The most basic one is lead taming, and it may well be the only one available to you at this point, the other three require some advancement before you can really start using them effectively, although all of them are somewhat useful at lower skill levels.
1. Lead taming has been discussed. Basically you make a macro for taming, bind it to a convenient key. Mine is bound to ctr-shft d. Most of my binds are on ctrl-shft, especially the ones I use most often, and they are all set up so I can access them easily with my left hand. Most of the "skills" in UO come down to key binds that are set up for easy access, and knowing them so well that using them is instinctive.
Once you have your key binds set up you just run up to the critter, and keep trying until you actually start taming - this can take a number of tries with aggressive creatures before you get passed simply pissing the critter off. The trick is to be able to use your taming macro and keep close enough to be able to start to tame and far enough away so the critter can't eat you. Once you have started the taming you don't need to stay quite as close and you can move a couple of extra tiles away.
One important thing with lead taming is having decent resists, especially if what you are trying to tame is a spell caster. If it isn't a spell caster, and you are a skilled lead tamer, it should never get close enough to hit you, if it is a spell caster having good resists is important, especially fire and energy resists. Sometime no matter how good you are, due to lag or whatever, a critter will get close enough to chew on you, in which case it is wise to have good resists that match the damage the critter does - cold and energy for cu sidhe, physical for hiryus, energy and physical for bake kitsune, etc. etc.
All of the other four forms of taming have limitations and you will not be able to use them in all situations, lead taming is the one that you can always fall back on, so it is important to know how to lead tame, even if you don't plan on making that your main method of taming.
2. Para taming is another common form of taming, however, for any of the more powerful tameables, you will need to have very high evaluate intelligence for it to work (for most of the high level tames 110 - 120). Also if there is any chance that the critter can have skills that will be over 100 after taming, you do not want to use it, as it will lower it's final skill cap in those skills (you can do a search on these forums for para taming and you should get more information on that). The current list of critters you do not want to para tame includes rune beetles, hiryu's (lesser and full), fire steeds, and reptalons. If the new dragon ever makes it to your shard in it's present form, not only will you not want to para tame it because of it's high skills, but it will laugh at you if you try...
With para taming you cast the spell paralyse, and start attempting to tame. Hopefully your taming attempts take hold before the paralysis wears off, if they don't, run off, cast invisibility on yourself, come back and try again. If your taming attempt takes hold you can then either lead tame it, or if your evaluate intelligence skill is extremely high, you may be able to cast invisibility on yourself before it is able to move.
When you become truly skilled at para taming, you will get to the place where you can tame most critters without even moving. For aggressive critters you cast paralyse, start the tame, and then what I do is cast invisibility roughly three times in a row, at the end of which I either have a tamed critter, or I cast paralyse again and start the process all over. With cu sidhe you only need to cast invisibility once as it is not normally aggressive.
3. Peace tamers are really the royalty of tamers and if actually taming critters is your primary object as a tamer, this is the template to choose. However for the more powerful tameables, you need to be a legendary bard for your barding skills to be useful (120 in both musicianship and peacemaking). With peace taming basically all you do is peace the critter and tame it. The peace may wear off part way through the attempt, in which case you will need to end up by lead taming (I'm not sure if you can use the peace skill again before you finish taming a critter, someone else will have to clarify this point).
4. Honour taming is so easy, it is almost criminal, but it takes a lot of work to get to the place where you can do this. Basically what you need to do is develop the honour virtue, and use that virtue on yourself before the taming attempt. Honour takes a lot of work to develop, and you lose a lot of honour every time you use it on yourself, so you can only use this method of taming sparingly. Basically when all of the other kinds of taming fail, this one is the one that you will be able to use.
You really need to have level two honour (Follower) for it to be truly useful. Level three honour (Knight) is extremely hard to come by, and even harder to hang on to, so I don't even bother with that. If you are an advanced tamer level two honour should be enough to allow you to tame just about anything.
To gain honour you invoke the virtue (again make a macro and bind this to a key - I have it bound to ctrl-alt-shft a because I've run out of convenient ctrl-shft key combinations for my left hand), and then you target some monster with high level fame before you attack it to kill it. This is called "honouring" your opponent. You can only honour undamaged monsters, so you must honour it before you attack it, and before it casts bless or strength on itself (this is called "buffing"). If it does buff itself before you can honour it you can curse it to return it to full health, if you have high evaluate intelligence, then attempt to honour it again).
The absolute best mob ("monster or beast") to gain honour on is the succubus. If you have a good cu sidhe they aren't too hard to kill, and they give a ton of fame, karma, and honour. However, if you are in Ilshenar, and a paragon spawns, be prepared for the fight of your life (in fact if you don't have a very advanced character and know how to use it effectively, you will die). However, if you do manage to kill a paragon succubus, I'm not sure that even peerless and Doom bosses give more fame, karma and honour than they do.
The reason why honour is so powerful is that if you honour yourself, no mobs will attack you. Even when you attempt to tame, you may anger your target, but it will still not attack you. However, you have to successfully tame your pet before the honour runs out, this is why level two honour is important. It is possible to tame critters with level one honour, but if the critter is at all obstinate, you will probably run out of time, and you won't be able to honour yourself again for another five minutes.
I am a mage tamer (para tamer), so I use honour on critters that you shouldn't para tame: rune beetles and hiryus primarily. What I do is kill all the ones that aren't good enough, and when I find one I like I lead it off to a safe area where it isn't surrounded by spawn, run off, cast invis, mark a rune, go stable my pet, come back and use honour to tame the beast. I'm very fussy about the pets I tame, but I still have to spend a fair bit of time hunting succubus for honour.
And this my young padawan, is the end of your lesson for today.