You put the toddler in a poncho, wearing sandals that can get wet, and you put a rain cover on your stroller. When you sit the toddler *in* the stroller seat, you pull the back of the poncho up and over the back of the seat so that you don't soak the upholstery too badly.
Rain covers are really not very expensive, and are a good investment b/c it often rains every single day in Florida for short periods in the summertime. You should NOT leave your stroller outside a ride without putting a rain cover over it, because when you come back 40 minutes later it may be soaked by a rain shower that you never saw coming and which is already over. Proper rain covers are best for two reasons: 1) they have air vents, and 2) because they are fitted it is easy to leave them on the stroller at all times and just gather up and clip them back over the hood when not in use.
If you don't want to buy a proper raincover, I recommend using a clear shower curtain liner with a few clothes pins or binder clips to hold it in place. Remember that it must hang loosely to allow for air circulation, and be sure to air it out thoroughly outdoors before you leave home; the fumes from new vinyl can make your child sick. Using an adult poncho works in a pinch, but the hood is apt to form a "bucket" right over the seat, that will often dump water into the seat as you try to move or remove the poncho.