How much milk will my baby need after he has started eating solids?

Variable. This is variable but a common question. Generally infants don't need more than 32 oz of milk per day and this will decrease depending on how much food the infant eats. At a year many infants are taking about 15-20 oz per day in my experience. Some infants will eat more food than others but all will naturally adjust milk intake dependent on food intake.
Varies with age. On average, babies start solids at 6 months. By 9 months, most babies are eating solids 3 times per day and taking milk 3-4 times. (6-8 ounces, usually) by 12 months, most babies take solids 4-5 times per day and milk 2-3 times (16-24 ounces on average).