Arguably the biggest mistake you can make is with "topping" your tree. It's considered something of a dirty word in horticulture circles. Topping means cutting a tree back so severely that only the stumps of large branches are left. It's done in an attempt to control the size of the tree. This stresses the tree because it takes away the leaves, its food sources. The tree will send out many smaller branches in an effort to produce more leaves quickly but they are not as strong as the previous branches and can break off easily in the wind. Tree topping also results in a pretty unattractive tree.
The way to properly bring a tree down in height is to make your cuts at a joint. A joint is the place where either a leaf or a branch grows out from the stem. Cut down the tallest branches at a joint, which will reduce the height significantly, instead of leaving a bunch of stubs. This way, the tree will be able to heal itself, growing a new branch in that spot.
If you have a dense tree that's completely blocking a view, you may want to thin it out. This involves removing entire branches that are too overgrown by making the proper cuts at a joint. This will allow you to see through the tree by taking some of the heaviness out of the canopy.
It's always a good idea to have a pruning partner whenever you prune, but this especially helps when thinning.
Before you cut, have someone stand back and look at the tree to make sure you're not chopping off an important part of the tree or taking so much as to ruin the shape.
You don't want to turn your pruning tools into weapons, so you better know how to use them. Pruners and pruning saws should always be sharp and disinfected when you use them. This will help you to make a smooth, clean cut and the tree will be able to recover faster from it.
For branches larger than 1-1/2" in diameter, you'll want to use a special technique called the "1,2,3 Method." This will protect the bark on your tree by preventing the branch from splitting. If you just started cutting downward on a larger branch, the weight of that branch would bring it down faster than you can cut. This would cause the branch to split and the bark would tear, making it harder for the tree to recover and more susceptible to pests or disease. Here is how this method is done:
Cut #1: Make an undercut about 1' from the trunk, which means cutting on the underside of the branch. Stop cutting when you've gone one third of the way through the branch.
Cut #2: Move about 3" past the undercut, toward the end of the branch, and start cutting. The weight of the branch will still take it down and the trunk may split but it won't split past your undercut.
Cut #3: Finish it off by cutting off the stub at the joint.