Branches growing straight up or inward are the ones to remove. Leave those branches that are growing outward. Side branches that have grown more than 1½ to 2 feet the previous year can be shortened. They should be pruned back to a side branch or bud that is facing the way you want the branch to grow. Generally outward-facing buds and branches are chosen.
Many times a number of vigorous vertical branches (called suckers or water sprouts) grow from the trunk or main branches after spring pruning. These are best removed (thinned) in June when shoots are less than a foot long. Extra vigorous side and top branches can also be shortened at the same time. Apple, pear, cherry, apricot and plum trees bear most of their fruit on short, twiggy branches called spurs. These should not be removed unless dead or broken. Of course some spurs will be removed with the excess branches that are removed.