Two acres is not a huge time expenditure but you do have to get a damn good mower. There are commercial mowers where you could also get a stand aboard platform for the 2 acres. I used to mow 5 acres of lawn with an old craftsman riding mower with no problemo. Wasn't flat so I didn't get bored, grins. Do not wimp out spending money on equipment. Never purchase electric. Stihl is one of the best brands for commercial/residential use. You should also own a gas blower and a gas trimmer (get rid of shield and go to a single strand replacement head). Forget the scythe (Ecnerwal you are so cute!!). You do not want the grass and weeds to get high enough where a scythe would work!!
SHARP BLADES! Keep those filters changed out!! Keep the dirt out of the engine and your equipment will last forever! Never ever use any gasoline with ethanol!! One mowing per week is MINIMUM for your lawn areas. Keep the lawn grasses no shorter than 3" and I am not kidding!! The pasture grasses will do fine with twice per season mowing depending on your expectations. Goats will be great. Pigs not so much (love pigs, brilliant animals but are able to dig up large areas of pasture...if just slightly unfed), chickens? Forget it. They will not eat the grasses or weeds. Perhaps eat the weed seeds and young starts but not good for maintenance. Goats are great. If you live next to BLM, or a greenway...BAD idea. Cougars, who are everywhere especially since we've been gobbling up territories from them LOVE GOATS! Another reason to never erect a playground next to greenways, native areas and pastures!
Grasses, not including domesticated lawn grasses, NEED to be cut down for health. A great weed wacker works well, 2 acres twice per year??? Make sure your riding lawn mower or your wide commercial deck can be adjusted...3" MINIMUM for your lawn (cool season grasses)...and I wouldn't cut less than 4-6" for pasture. The HIGHER your cut, the higher you leave the grass blades the SLOWER the growth. Too low and you are cutting off photosynthetic growth necessary to support the roots and you'll wig your grasses out big time.
Don't forget! Your lawn needs at least 3X or 4X fertilizations that are seasonally made compositions. Your pasture needs at least ONE great fertilizing. Don't use fast acting Scott's, Ortho, or any big chemical company. The best fertilizer I have ever used is ORGANIC (I hate using that word), SLOW RELEASE (intrinsic), with bacteria and micronutrients not found in the FAST STUFF. After decades of fertilizing grasses I was just blown away at the difference. Here is one and you can look for similar brands...Dr. Earth Lawn Fertilizer. Make sure the last fertilizing before winter is LOW NITROGEN.
When you aerate your pretty lawn, aerate your pasture (once per year minimum). Only use the aerators that pull plugs of sod and soil out...leave to disintegrate on your lawn/pasture. If you have goats to take care of the pasture...they might be a bit tough on eating the grasses down too far and one needs to cut the fertilizer way back. Include dividing your pasture into sections so once one section has been eaten down, put the goats in the other. This allows your pasture to come back, a good time to reseed.