Making the Most of Your Time
If seeing both the Gulf and Atlantic coasts is a priority, then you really only have one choice and that is to head west (southwest) from Orlando. I'd actually bypass the Tampa Bay area and head directly for Port Charlotte and then follow the coast south, taking US-41 rather than I-75 from Naples onward. Even though the main entrance into Everglades National Park is from Homestead, there is a less utilized entrance on the north side, Shark Valley, off US-41 and I'd recommend that you try them both on your way down to Key West. You'd then get the Atlantic coast in Miami.
AZBuck