It's cold and sometimes rainy, a perfect time of year to make a nice cup of tea and wrap up in a warm blanket.
It's also the perfect time of year to get some needed pruning done. The trees are dormant and for us in Arlington we have pretty nice weather.
There are three reasons to prune a tree, shaping, fruit production and removing damaged or dead material.
If you want to shape a tree you first need to figure out what shape you want the tree to take on. This will depend on what kind of tree it is and where it was planted. An apple tree for instance usually need plenty of room to spread out and you would thus prune the branches to give plenty of room between the main limbs. If you moved into a house where the apple tree is in a more cramped space you might do something similar but with the branches moving more upward, cutting off branches that might want to be hitting the house or other structures.
For fruit production you will want to remove sucker branches. These grow more or less straight up from the trunk or sometimes large branches. they usually don't produce fruit but are easy to remove since they are new and green. You would remove these in the spring when they form. They can also be removed in the winter but by this time will have become woody. If you bought a bush from the store then the rootstock might not be the same as the rest of the bush. Suckers originating from the base may not even be of the same type and can take over if not removed.
For removing dead or damaged limbs you will need to figure out what limbs need removing and make sure you have removed enough. A rotted limb might go into a larger limb and the entire thing might need removing. This happened with my fig tree. A limb as big around as my arm had to be cut out. The next year it produced better than ever since it didn't have to work on healing as much.
One of the main things to consider with any pruning is making sure the tree as a whole is not damaged in the process. A cut too far back can damage the limb or trunk the pruned limb is being cut off of. Better in that case not to cut too much lest you have to cut the next branch too.
I would recommend cutting these off as soon as you see them start to grow.