Llamas don't like to be caught but once you have them they usually give in and are easy to lead and work with. The following steps are for a llama that has been caught before and is lead broke.
You need a small catch pen. Ten by ten is best but a long narrow alley will also work. It needs to be narrow enough that you can move back and forth to keep them from going past you.
Most llamas will not brush by you if they think they are going to touch you when they do it. This really works to your advantage. You need to remain quiet and not hurry things as that gets the llama upset and my advice won't work. Then they will push past you and you will have to start over trying to calm them down.
Keep in mind that llamas are creatures of habit and they will usually react the same every time. You will find that they have one certain place where they "give in" to you catching them. They will also react the same each time when you try to put the halter on them. They may stick their face through the fence or stick it in the air or some other avoidance method. Learn what each of your llamas likes to do and work with them on it.
Get them used to going in the catch pen by feeding them in it. Do this so much that they are comfortable there. Don't just try and get them in it when you want to catch them. They will quickly figure this out and without the catch pen you won't get them caught. (It is a very rare llama that you can walk up to in the field and catch...very rare.)
Once in the pen, go in slowly but with authority. Don't act like you are afraid of them. Stand straight and move slowly but without hesitation. Don't walk straight toward their head. Instead aim for their shoulder or side. This seems to be less intimidating to them. Hold your arms out to the side to make a "person fence" to keep them where you need them. Most won't rush your arms but try to avoid them. Just don't YOU rush them.
Llamas learn vocabulary. Use the same word each time and with the same tone of voice. I use "Stand". When the llama stops moving, you stop too. This is their reward for doing as asked. Wait a moment and step toward the center of their back. Say "stand" again and when they stop you repeat your stop and keep doing this over and over. Talk calmly to them as you stand there. Watch for them to relax a bit before you move again.
When you are close enough you may be able to touch them on the back or put your arm around the neck. Do this with a firm feel. Llamas do not like a light touch. Some llamas give in when you rest your hand on the back, others don't. You have to learn the individual animal.
Some llamas give in if you put the lead rope over their back. If you can get it over the back up by the neck, try and keep it there even if they move around the pen. If you can get it around their neck, try and move it up toward the head as this gives you more control and leverage.
If they move away you will have to repeat the process until they give in. Remember that the reward for the llama standing still is that you stop heading for them for a moment. If they try to bolt and run by you, you will need to move in front of them to block them. Do this until they stop, then go back to saying "stand" again.
Once you can get close enough to slide your arm around their neck, you may have to move with them as they try to get away. Now is not the time to wimp out...go with them. Most will stop soon especially if you say "stand".
I say "Halter" to let them know I am going to snap the lead on them or put the halter on their head. They like to know or be warned as to what you are going to do to them. So get used to verbal commands.
At this point you should be able to snap on the lead. If you need to put the halter on this may take more doing.
To Put the Halter On and Off
Try and keep them under control by having the lead rope around the neck up by the head. Hold it in one of your hands along with the halter. That way if they try to back away you can keep them with you.
If you are able, it helps to position them so their rear end is in a corner. This keeps them from backing away from you. Some llamas however, will not let you move them until haltered so you just have to work with what you have.
With your right arm around the llama's neck, use both hands to get the halter ready. Hold it below the llamas face. They hate things coming at them from above so always put it on from below.
Have the nose piece open and lift it over the nose. Try not to slide it on the nose but lift it over onto the face just below the eyes. Then fasten it. Halters with stiff nose bands and a snap instead of a buckle are the easiest to use. They are worth the investment.
When you take off the halter, use the neck rope to keep them from getting into the habit of pulling away as you take the halter off. This is a bad habit and one they learn way too fast so don't let them get away with it. Some llamas already have learned this trick. With these llamas, turn them so that their back end is in a corner. This way they have no place to go should they try to bolt backwards.
Be sure not to slide the halter off their nose. Instead lift the nose band up and lift it off. In time you will find that your llama no longer tries to bolt away from you and may even stick their nose into the nose piece when you hold the halter out in front of them.
Llamas learn fast. Unfortunately they pick up bad habits just as quickly. Try to be consistent in what you do and the commands you give. Work with their quirks instead of fighting them to do it your way. If they want to stick their head through a fence, you need to learn how to put on the halter with it there. If they tend to spit when you approach, just get ready for it and ignore the behavior. Most won't direct the spit at you. They are just showing their displeasure or fear. Don't reward them by reacting to it.
Don't rush the process. If you are getting upset, stop and come back at another time.
Once you have the llama caught, do something fun as a reward. Leading them around to see the sights or to a place where they can eat some green grass or fresh hay will make catching them much easier the next time. If you need to trim feet or give shots, do it after they have had their reward and settled down.
Working with llamas is fun but you have to catch them first. Have a small pen, lots of patience and a positive constant demeanor and you will do just fine.