Buttercups grow where there is a lack of lime on the land. Lime will kill them, but you can also get a weedkiller to do the same job (without your goats in the field preferably)
Why do you think the grass needs improving? Is it meadow grass (full of herbs, different grasses, meadow flowers, etc - my favourite!) or is it just rough land, or ryegrass, which is grown for horses? You can put fertiliser on it, but that will spoil meadow grass if there is any.
Mowing the grass will get rid of the old grass, and encourage new growth, but it will not change the quality of the grass really. At this time of year, until the end of June, you should be getting good grass whatever the type. Sheep would eat the grass right down, as they will eat grass goats leave behind.....goats will only eat the bits they want.
Remember, goats are browsers not grazers, and if given the choice would eat leaves over grass. I am out every day with a builders sack collecting enough to keep all my gang happy - they love the leaves at the moment as they are new.