When my daughter turned 1 year, our pediatrician suggested that she get no more than 16 oz of milk per day to be sure, as one of the other commenters suggested, that she doesn't get too full and replace other nutrient-rich foods with milk. And on the other end of the spectrum, he said not to be too concerned if she doesn't drink much because he said she really doesn't need milk at all. Humans are the only mammal that continue to drink milk once weaned from the breast. Just be sure to provide the nutrients in other ways - i.e. yogurt, cheese, and other dairy foods if you feel he is not getting enough milk.
I offer my daughter milk with meals and water with snacks. In the beginning, she didn't drink much but over time she has begun to drink more. As for juice, I think there are many different opinions on this, but I don't give my daughter any juice and don't intend to until she's 18 months or older. So many fruit juices have added sugar, and my daughter eats plenty of fruit during the day so I don't feel the need to give her any juice.
Another thing you may consider is the sippy cup you're using. My daughter didn't respond very well to the first sippy cup we gave her. It had a soft spout and she tended to chew on it rather than bite and suck, so we switched to the First Years Take and Toss cups which have a hard spout and are easy to sip (at least for my daughter), and she has taken to it very well.