Finding a rental property for your needs will have a lot to do with your credit, past rental history, and other terms that the landlord requires. Some will allow pets with a deposit for each animal. If the animal causes damage to the property the landlord will have some remedies including, but not limited to, keeping the deposit(s). Other questions to consider are where you want to be (school district, etc), how many bedrooms and baths, square footage, and of course your budget. Be sure to remember to include utilities into the budget. Often you can locate a rental by driving around the areas you like and watching for rental signs.
There may be some properties on the Tyler MLS system that fit your criteria. Contact a local real estate agent for this information.