Can I cross private property to get to a section of national forest?

In a word NO. Unless you have said land owners permission. Many land owners will allow you to cross their land if you ask nicely and explain your purpose for wanting to cross their land. The other possibility is if a piece of National Forest is land locked (boardered on all sides by private land) Some of the private land may have what's called an easment which is an access point to the National Forest.