I have just moved into a home with a peach tree that appears healthy, but overgrown; it has clearly not been pruned in many years (although I have cut out some of the branches that made it impossible to walk under the tree).
How can I help this tree reach its peach-producing potential?
The tree is on the south side of the house; photo looks northwest.
Edit: after the information from David that the tree produced heavily this year, renovation may be a practical course of action.
You may get a better answer from someone else, but I'd suggest that you should cut it down and plant a new tree. Peach trees are not long-lived -- a 20-25 year old tree is ready for retirement. Production typically drops off at that point. You might be able to prune this over a couple of years and have it produce some fruit, but I find it unlikely that you'll be paid back for your time and effort.
Even if you could renovate the old tree and get a few more years of crops from it, a new tree will begin producing within a year or two, will have a longer productive life, and you would expect larger crops than the old tree within just a few years.
See this article on fruit tree pruning, especially:
You will seldom have to resort to serious renovation pruning on peaches, nectarines, sweet cherries and Japanese plums. These and most apricots seldom live long enough to need renovation. A peach at 25 is old.
A major renovation will take a couple of years -- you don't want to remove too much at once. Start by removing dead/diseased branches and then stop. If that doesn't remove much material, remove branches that are crossing, growing the wrong direction, etc. But stop before you remove too much. Then take a look at what happens next year.
Keep in mind that fruit forms on this year's wood. So if you want fruit next year, make sure you leave some of this year's growth.
If you have the tools, and you can do the job safely, and you want to reduce the height (for ease of picking and/or more pleasing shape): in the second year, you may consider removing some of the upper branches. But it would really be worth calling a local arborist with experience handling fruit trees to do this kind of work -- both you and the tree may end up healthier.
Normally you'd prune while the tree is dormant (in winter), but you may want to think about a summer pruning since heavy pruning during winter dormancy can stimulate vigorous growth at the expense of fruit production. And vigorous new growth isn't what you want.
Also, if you do figure out how to keep the squirrels away, make sure you thin the fruit to avoid breaking branches.
I'm definitely not going to contradict bstpierre's answer, as I'm far from a tree expert, but if you do decide to give the tree another chance, you will find the below tree pruning document/guide helpful:
How to Prune Trees by US Forest Service
Also, "Tree Care Information" from Trees Our Good contains some useful information on taking care of trees.
