Ah, childhood memories come flooding back...
I was my mothers chief goatherd (and shepherd) up to the age of 19 when I finally escaped to go to college.
She had up to 9 at a time, including Saanans (classic white goat), British Toggenburgs (brown and white jobs), British Alpines (black and white) and Anglo Nubians (African stylee goat, floppy ears and big noses)
We kept them for milking. On the plus side goats are total characters, independant (bloody) minded and wilfull - much more entertaining than sheep. If you breed from them, the kids are cute as cute can be. Another advantage is that generally they're pretty hardy.
On the downside, you will need bullet proof fencing (they are experts at eating/barging holes in most hedges), a decent shelter (they hate the wet) and provision for all the regular issues that come with any livestock - trimming hooves, worming, mucking them out occasionally, trips to the vet for inexplicable intestinal complaints etc etc. Don't ever let them anywhere near a garden or any area with shrubs or trees you value - they will trash it at a speed you will not believe.
Your 3/4 acre will probably support a pair (and they're happier in gangs) so long as you supplement their fodder with adequate hay etc in winter.
Unfortunately, they probably won't be massively effective at keeping the grass down - they are as much browsers as grazers - sheep would be more effective at keeping the grass short. Although sheep come with a larger bunch of issues re dealing with worms, flystrike, shearing etc.
On the whole, I'd say goats will give you a better chance of something that bridges the gap between agricultural animal and pet, but sheep might be more functional for the lawnmowing. Or what about geese?
HTH, Gizmo
PS don't go for the Anglo Nubians, by the way - they jump like deer!