Lots of choices
A lot depends on whether you're going to have your family with you while birding. If so, Brazos Bend State Park is your best bet. It's at most a 45-minute drive from downtown, there are plenty of big birds (e.g., herons, egrets, anhingas, whistling ducks) that kids can enjoy, plus lots of alligators in April. It's okay but not great for other birds. Walk around 40 Acre Lake and then park again at Elm Lake and walk around that. The swampy areas beside the trails are very different landscape from anything you get in the UK. There's a small nature center with snakes in cages, soft drinks and sweets.
San Bernard NWR is 30-40 minutes further away but is much better for a range of birds. It's mainly an auto-tour with stops plus one nice boardwalk through a swampy area. It's not so much fun for nonbirders.
If you're on your own, a good trip would be a loop. Start with a 90-minute drive to High Island for migrants. (Skip Anahuac NWR because of damage by Huiiricane Ike.) Continue 30 minutes to Bolivar Beach for seaside birds; stop along the way at Rollover Pass for more seaside birds. Take the free ferry to Galveston and then drive west off the island to spend the afternoon at Brazoria NWR or San Bernard NWR or Brazos Bend State Park. It's a lot of driving but you'll see a lot of birds.
I've never heard of crime-related safety being an issue for birders in our area, though I guess it could be in certain areas of Houston itself. The driving is easy, except in rush hours on weekdays. Mosquitoes will be getting active so a strong repellent is advisable - 20% plus of DEET. You also need to watch where you put your feet: snakes shouldn't be much of an issue in April but fire ants and chiggers are very nasty.
BTW, I've seen other Brits encouraging their kids to go right up to alligators on paths at e.g., Brazos Bend. (Like those idiots who used to get out to picnic in the lion enclosure at Windsor Safari Park.) Not a good idea! The alligators are not tame.
If you look through my blog (
www.jeffincypress.blogspot.com), you'll see photos and reports from the places I've mentioned.
Jeff