Good for you for waiting to prune your tree! Summer is the worst time to remove any foliage.

Suckers are shoots that emerge from the base of the plant/tree and yes, you should remove them, even in summer. When they are new and tender you can just use your gloved hand to brush/rub them off. If they are older and tougher you'll need to use bypass hand pruners.

If shoots emerge along the trunk of your tree it's OK to leave them for about a year. Your newly planted trees will need as many leaves as possible to act as food factories for the tree and to provide a little shade for the trunk.

It's not necessary to prune the tops of your privets now. Have patience and wait for about a year before you begin to direct the growth by pruning. Again, the leaves are food factories for you newly planted shrub or tree. Depending on your purpose for planting privet you can selectively prune either into a small tree or hedge. If your goal is a small tree, begin removing lower shoots next year to 'train' it into a tree shape. If you want a hedge-type plant remove upper shoots at a node next year which will cause multiple branches to grow at that point.

A good reference on pruning is Pruning, Planting and Care by Eric Johnson - Ironwood Press. It's written for our region in the Southwest and is available on-line or at most local bookstores

Great, thanks for your help. I ordered the book and should be receiving it shortly.