I have used these people before - they also sell butterworms too, which I had forgotten about!
http://www.butterworms.co.uk/
With the silkworms, you just need to keep them clean and feed them and when they get large enough they will spin a silk cocoon and pupate inside it. After 7 - 10 days the moths begin to emerge and will mate pretty much straight away if there is a pair. They will stay locked together for over 24 hours and then they will part and the female may rest before laying her eggs. Some rest and some just get on with it. Once the eggs are laid, they turn dark grey/purple after about 3 days, then you can put them into diapause in the fridge (this simulates winter for them) for at least 4 weeks, then take them out and incubated them and start over again!