I live in Leicestershire and our local is the Atherstone. I don't actually hunt but I know a fair few people who do and they all say how friendly it is. Can't help with prices though - sorry.
PS whereabouts in Leics are you moving to?
Just look at it as 'you pay for what you get' ie you'll pay more for better country and better jumping. Phone round the secretaries explaining that you're moving to the area, and enquire about prices and such like.
Having hunted in Leics with the Fernie, Cottesmore and Quorn I would say your best bet would be to go to the Quorn on a Monday or Friday. Its absolutely fantastic with tons of hedges and seriously fast hunting. I hunted with the Cottesmore as my local pack and they are good fun too, again with lots of hedges. Best day is a Tuesday. Sats and Thurs are a bit dull. The Fernie has a lot of very manicured hedges and hunt jumps and lovely country. If you were Harborough way then you'd want to be with the Fernie. You could join the Harborough Hunt Club or the Melton Hunt Club which means you can hunt with all 4 (I think) packs for very little - 2 days each and do the hunt rides which are brilliant fun.
All three have lots and lots of grass!! The Belvoir who are more over towards Lincs have the same kind of jumps but heavier going and quite a bit of plough.