Generally if you trim it's in the area's where they get dirtiest- their bums, the bits at the side of their bums etc. One thing to note is they sometimes grow hair on their chest in between their legs- its best to trim this to make them look neater. Also, normally trim beards as well.
I have seen goat's completely clipped- but generally it tends to be goats that have proven themselves to be bad at getting rid of their winter coats. If it's just your two goatlings you have, it is unlikely they will have any hugely dirty bits that need trimmed. It's once they turn into heavy milkers, that they tend to be a lot messier, and they will make a point out of finding a horrid bit of poo, sitting in it, and somehow grinding it in! If they are constantly doing that, they can become stained, yellowy brown, and it really is best to trim/clip that.
If you clip completely- it takes 4-5 weeks to grow back, so do it in plenty of time before the show!
Beth