As a dairy farmer, I would recommend a dairy cow for the purposes you listed.
I have holsteins, jerseys and normandes.
The holsteins are one of my favorites and have very sweet temperments. But if you are only going through a few gallons per week, then they will give WAY more than you will use!
The jerseys are great, much smaller, VERY friendly, but tend to be more bossy. Some of the ones I have kick like a mother too! So if you go the jersey route, I would suggest working with her a lot to get her used to being touched (if she's not already a milker). And they do have a noticably higher butterfat, but with the amount you say you'll be using (for butter) you will have plenty of fat with a holstein as well. The jerseys will also give less milk, but they can still produce much more than what you need!
The normades are right in the middle. I have some that produce as well as my holsteins, and some with butterfat as high as the jerseys! And they are a shorter, more stocky built cow that is EXCELLENT for beef too. If raised properly, they will have nice marbling. You can eat the jerseys and holsteins, but you will find them to be more lean.
My recommendation would be to visit a local dairy farm and talk to them about the cows they have. And raising cows in general. A lot of people don't realize how much work/feed/etc goes into a cow until they get one and see for themselves. And cows are not seasonal like goats and sheep. They need to be milked at least 2 times a day until they aren't producing much (they can be high producers for up to a year). And then you might decide a few goats are better for your needs! So it is always best to research and find out as much as possible before you make a decision.