Good morning Ms McKenna, 

Thank you for reaching out to the UNH Extension Infoline. As a general rule you prune younger trees lightly because you want them to "make wood", to grow bigger, but some pruning can be done early on to create the open center that you want on a peach tree, no tall single stem, but strong lateral branches going out to the sides to bear the fruit. You say your tree is "new" so I have the impression that when you say the tree is "new" that the dozen or so peaches were your first harvest. that means you can do some after first harvest pruning. I am attaching links to papers on pruning from Clemson, Penn State and UNH that should give you some good guidance about how and when to prune. In addition, UNH has pruning workshops every year where you can go to an orchard, and learn how the experts do it. They are usually in late winter or early spring. When you prune, you should have good quality tools, including hand loppers and a small pruning saw to start.   And I will add one personal aside--beware You tube videos unless they come from a university extension service.

https://hgic.clemson.edu/factsheet/pruning-peaches-nectarines/

https://extension.unh.edu/blog/2018/02/why-we-prune-fruiting-plants-when-do-it

https://extension.psu.edu/peach-tree-pruning-managing-light-and-crop-load

To check on workshops go to https://extension.unh.edu/  then click on agriculture and gardens.