I had a dairy, raised beef cows and horses in Tillamook, Oregon.
Any cow can produce milk and be trained to milk. The reason there are beef cows and dairy cows is some breeds produce more milk. Others produce a higher butterfat percentage. Some make meat faster. Also dairy cows produce bigger calves and about 10% of the births need help. Beef cows have smaller calves and are bred to have calves on their own. A cow has to be bred once a year and have a calf to continue producing milk. They need to be dried off (stop milking) 60 days before the calf comes to rebuild their body resources. The milk volume is greatest about 30 days after calving and gradually decreases. Grain and good hay increases output, but once output drops (from poor feed, missed milkings, injury) it's difficult to bring it back until the next calf.
The easiest method is buy a cow that is already trained to milk. Raising a calf to milk is better than a grown cow. Think of a grown cow as almost wild. First you have to tame it, then train for milking. When training a cow, kindness and persistence gets them trained. They should be brought into the milking area several times before calving to become comfortable there. Dairy associations should have lots of information. You need to think about how you're going to milk. By hand or machine. Hand milking requires strong hands and fingers. Arthritis is common in old hand milkers. An old milker I knew had to soak his hands in ice water before doing anything. Hand milking takes much longer than machine milking. Some cows may be more trouble with extended milking time. Many udder problems come from poor milking practices. Read up on it. An infection in the udder is difficult to cure.
I don't know about smaller breeds, but a Holstein produces about 100 pounds of poop a day, not counting urine. In a commercial heard they consume 30+ pounds of grain, 50+ pounds of alfalfa, and all the pasture they can eat depending on milk production. What they eat affects the taste of the milk. Exotic foods usually are not worth it.
If there are livestock sales near you, dairy cows, no longer producing enough milk for commercial dairies, can be bought for less than the cost of beef. Usually they will have one or more quarters that can no longer be milked because of injury or infection.