Thank you for contacting Ask Extension. What you have is not an Elm but a Callery Pear Tree, or a Bradford Pear, is the common name. They are considered very invasive in the midwest. 

You can trim crossing branches and waterspouts anytime, but to do any significant pruning, you should do late spring so that the tree can harden up for the winter. Never remove more than 1/3 of the tree branches at any time. If you would like to talk with an arborist, you can go to http://www.treesaregood.org to find one near you.

Here are some links for more information:

https://extension.psu.edu/callery-pear

https://www.pubs.ext.vt.edu/430/430-456/430-456.html