For pets/weed control your best bet is probably two wethers. They are easy, you can usually get quality ones for a great price, and require less maintenance a doe. You don't want bucks for pets. Breed is probably more of a personal choice and isn't quite as important if they are just pets, but some breeds have pros/cons. I would highly suggest either a naturally polled goat (not super common) or one that has been disbudded. You can call any local dairy goat breeders and ask them about pet wethers. Most have "extra" ones around and they are generally disbudded, bottle raised, and friendly. Around here it's easy to get one for $50 or less.
My daughter has a registered LaMancha doeling (for breeding, milk) and a boer-mix wethers to keep her company. The doeling was bottle raised and the little boy was dam-raised and they are both very friendly and great with our human kids. They never butt people, they run together, etc. She wants to train him to pull a cart when he's bigger. Ours live in our one acre fenced "backyard" but have a pen with a shelter for when we need them out of the way. They will climb on everything, so be prepared. We have five acres all together and they go out there supervised, but the fences aren't "goat proof."
As far as "mowing" they do eat grass, but they also love weeds, vines, branches, etc since they are browsers, so depending on the size of the area you may find you still need to mow sometimes. Also, they may not eat the grass when it's really wet because they don't like standing in water at all. They will go in once the ground dries out.
