Hey B.,
I just actually trimmed all our own trees in our backyard because of the too much shade issue and wanting to lay sod. The thing to remember when trimming big branches is to not cut too near the trunk. You want to leave a couple of inches of branch so that you're not hurting the main system. I also found that taking down the branches in sections really helped, working from the outside in, especially on the larger, heavier branches.
Since your trimming to increase the sunlight, go out underneath your tree and take a good look at the main shade branches to figure out exactly which one's you can cut that will allow the most light through. In taking 2 large branches off my own tree, I increased my sunlight into my yard by a good few morning hours. Once you have cut one off, stop and take a step back (across the street would be best) to see how much that changed it. Really try to use discretion about the number of branches you take.
St. Augustine is a good choice, but remember that it still needs a good 3 hours of sunlight to really survive, so if your tree is too dense, I second the idea of planting a low ground cover around the base of the tree, with the grass planted further out. Other ideas would be elephant ears, hostas, caladiums, etc. Just be sure to till up your soil, till in some good fertilizer, and rake it smooth before you lay your sod. Also, the first two weeks you've laid it are crucial. The sod needs to stay moist, but not in standing water. St. Augustine is prone to fungus infections, but with the weather warming up the way it is, this probably won't be a problem for you, unless you have a heavy clay soil.
My husband was too busy to trim our trees, either, and the task of trimming them by myself definitely seemed a little daunting at first, but once you cut off your first branch and see the sunlight start poking through, you won't want to quit! Just remember to wear some kind of eye protection (I just wore my sunglasses), because all that sawdust will fall right down on you. And remember to stand behind the cut - meaning away from the path of the branch falling.
Good luck girl! You'll be glad you did all the hard work once you have your lawn back.
E.