That's sounds like an awesome idea! Goats are so much fun, and really not that much work, but you'll probably find yourself wanting to spend more time with them anyways.
We use to keep goats with our chickens (until the economy got really bad), and it took about the same amount of time to care for them. You let the chickens out of their secure night pen and the goats too (we had mini goats and didn't want coyotes getting them). We would give them buckets of water (they shared), the chickens were offered food in their room and the goats got some hay and grain. The door to the pasture was opened and that was that.
However, if you really want to get milk from a goat you'll have to spend more time making sure they are tame, have the proper nutrition to produce milk, and that they are milked twice a day at the same time, which can take extra effort.
Just like chickens, goats will need their pens cleaned as needed, which means an extra half hour or so once every week or two. They'll need their water buckets scrubbed, hooves trimmed (as needed), dewormed, and given health inspections to look for any signs of problems (the earlier they are caught the better). Not to mention I've found goats to adore any time they can spend with people, getting scratched, walked, and treats have got to be on the top of their list. They'll steal whatever they can from the chickens (and cats for that matter) and try to break into the feed room at any chance they get as well. However, the only real danger they had towards the chickens was with trampling them, which to my knowledge only happened once, and it was with a chick.
I hope this helps some. Best of luck!