Answer: The only way to counteract the wetness in the lawn is to install drain tiles or some kind of drainage system to give the excess water a place to go. As for planting, you don't want to work overly wet soil; give it a chance to dry out before digging and then look for plants that will take either sun or shade (depending upon the conditions in your landscape). Just cruising through the garden center at your local Home Depot will give you tons of ideas. While you're there, check with the staff for recommendations on drying out your lawn. They can show you how to install a curtain drain (it's fairly simple) which will help your lawn dry out faster in the spring.
Best wishes with your landscape!
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