I actually live in Greatwood and love it! There are 3 daycares in Greatwood....a Kindercare (where my daughter currently attends), a Primrose (which I didn't like the staff there when I visited), and a Montessori school (where I think I'm going to switch my daughter to once she is old enough to go there (18 months)....I haven't visited this school yet, so I will need to do that before I move her as well).
As far as activities....there is a Little Gym and a Gymboree in Missouri City which is a good 15 or 20 minute drive from Greatwood. My daughter and I go to a Gymboree class only because their schedule fit our schedule better (I work 3 days a week). I know that libraries in the area offer toddler times.....but they are all in the middle of the week so that is out for us, but since you don't work it could be a great thing to do with your kids! I have heard that kids really enjoy it.
Greatwood has 2 or 3 different pools and many playgrounds for kids to play on. I don't know anything about the Sweetwater area neighborhoods, so I can't help you there.
And if you have never lived in Texas....just get ready for the heat and humidity!!!! : ) I'm sure you'll love it here.
Good luck in your move.