You can get all this information from the TPW publication Chris referenced, but I'll give you a few highlights.
Deer hunting does tend to be expensive, if you don't have land or know someone who has land you can hunt on. If cost/value of your meat is a consideration, you are probably not going to want to pay for a lease. The cost of the lease is going to make the cost per pound of your meat pretty high. You do need a hunting license to take deer even on private land. Some counties may have minimum property sizes on which you can hunt, but in most cases I think 15 acres is going to work.
Wild boar are, by a wide margin, your best bet for hunting if your purpose is to provide food, and cheaply. Hogs run rampant, and do a huge amount of damage to crops and livestock. There is no closed season for wild boar (javalina are a different story), and they can be taken with any method you choose. Some people have packs of dogs that will pin the pigs down, then they run up and stick the pig in the neck with a knife. Many farmers will be happy to let you trap pigs on their land, some may let you hunt with other methods. They can be hunted day or night. If the animal isn't too big, the meat is terrific. The best bet is to trap them, then feed them on corn for a few weeks before slaughtering them. Makes the meat better.
I don't know many people who hunt them, but rabbits and squirrels can be hunted year-round in many counties.
Dove hunting is wildly popular, at least in central Texas, though it isn't a way to get many calories. Mourning and white tip dove have a limited season, pigeons are open all year. In the past there has been a program which allowed you access to a great deal of private property for hunting for about $40. Usually restricted to dove hunting. I can't find the information on it for this year, I hope that they have not discontinued the program.
Waterfowl are not as popular where I live, but then, I don't live in a place with a lot of water. Probably more activity in other parts of the state.
Generally non-native species (such as wild boar and many imported ungulates) can be hunted any time. However, with the exception of wild boar, you aren't going to find a lot of that.
Coyotes can be hunted any time. I think that they have to be legitimate "pests" (did damage, are damaging, or are threatening to damage crops, livestock, or other property). But if you see a coyote, it's doing something it shouldn't be. Like eating my entire flock of chickens. Unfortunately, I don't think coyotes provide prime steaks.
Other pests can generally be hunted any time as well, but most of them aren't good eating. Bear cannot be hunted in Texas. Mountain lions are considered pests, and can be hunted at any time.
