Question
I have 2 patio peach trees and a weeping crabapple tree.  In trying to determine if and when I should prune them, I discovered that they are "grafted" trees.  Because of this, I now understand that shoots (ie., suckers) from the root stock should be removed as they appear.

Since I was not aware of this, I now have new trees growing adjacent to the original trees.  From what I am reading, I suppose I should remove these as soon as the flowering stops, but before June.  I would appreciate your advice as follows.

1. When and how should these be removed?
2. Is there anything I can do with what is removed, like replant it?  Or, should it just be discarded?
3. As new shoots appear, should I dig beneath the ground and remove them from the trunk?  

Answer
Yes, remove the suckers from the trunk and discard them. They are sucking energy from your graphed tree. These shoots are from a plain root stock that is used for it's good qualities, but not for good fruit.
The part that is graphed on to the root stock is the one that you need to take care of. If you don't get rid of the suckers right away, it will weaken the tree, leaving it vulnerable to disease and insects.
If they are large you will need a saw to cut them out, otherwise you can use a shape pair of loppers. Try to cut as close to the tree as you can without damaging the bark.
The main pruning of fruit trees should be done in early spring when the new growth starts.
If you live in a cold climate, it is best to wait until after the daffodils bloom. The new buds that come after pruning, will need protection until there is no more frost. Peach trees are semi-hardy.
The crabapple is hardy.
When you prune, take out any dead limbs. Take out any limbs that are growing toward the center, or crossing. Then, look at the tree carefully to see if any thinning needs to be done. Look for limbs that a spindly. Cut these just above the leaf node. Remember that what ever way that node is facing on the limb, that is the way the new growth will go. So, visualize how you want the tree to grow, as you shape it. Don't over prune your trees, you will sacrifice fruiting.
Take your time, and do only what is needed.