It's best to hold off a while before actually putting in the tags until the lambs ears are strong enough to take them - as long as you can identify each lamb before that. We use turkey wing tags at birth as temporary ID of new lambs then put in the permanent ones once they are sold on at 4 months, or 9 months for those we keep.
We use Allflex bubblegum eid tags which are generally accepted as good - they are the dangly type, but soft, with a good pin, plenty of room for the developing ear and clearly marked. We have in the past used button tags, before they had to have eids in, and we found them ok in general except that they tended to get rolled over ears caught in them on application, in breeds with small ears. Some other users hate them. Look carefully at the tags you choose to make sure there will be room for the ear to grow - some tags get narrower towards the folded-over end and have caused problems, others don't have enough room at the pin end for thick ears.
You can order all your tags online or by phone - much quicker than going through an agricultural store which just act as agents and take up to a month. Defra has a list of recommended suppliers - these suppliers will take the details they need from you and supply legally appropriate tags. Try googling Allflex for the bubblegum ones - they will send out samples. They also do a slaughter tag which has a very fine pin but if you want them to stay in for more than a year or so would probably not be suitable for the long term.
The best way to ensure that your eartags stay in and don't cause infection is to apply them properly in precisely the right place between the rays of cartillage in the ears. We have been putting ours in for the last few years with the fold at the top of the ear and we have found these catch less often.