Hi Maddy - maybe needs a long answer... just about to go out and do the animals...
You can AI goats either DIY (but will still need semen straws stored in liquid Nitrogen) or use Innovis who do Laprascopic AI. I am about to do one of my GG's at Innovis this year.
Straws of the various breeds are kept by lots of different people and finding out about them is not always straightforward, but most of the breed societies will know what's where - what breed are you thinking of doing?
For laprascopic goats are sponged, and then injected with PMSG when sponge comes out to make sure she is at the ríght stage when they do it. From what I have heard it is done quickly and efficiently. It s very similar to sheep - I think Tim Tyne wrote an article in CountrySmallholding a wee while back.
Haven't added up the costs for mine yet - the most expensive bit is to get the straws transported and stored somewhere. Best thing is though to be able to use a male long dead, but in your pedigree...
If you plan to do it yourself (or actually use an inseminator - BGS has got list of people having attended the course run by Christine Ball of GoatGenetics) you would also need to locate a tank for your straws.
Where in the country are you? Check out Innovis's website, the bit about sheep AI is what you need.
You are maybe a bit late to get it organised for this year though... getting the straws to where you want them can take a while... I have been doing this for most of the summer...