I don't know what part of town you will be looking to move to, but if you have a Realtor that really knows the area and works with a lot of property management companies, he/she will be in a better position to negotiate for you.
Welcome to Houston!
Make sure that the rental form is the current, standard, TREC form and that there are no unusual charges or notations is the special provisions section. You will be required to fill out an application for each person residing in the premises that is 18yrs or older.
Check to make sure that the property conforms to guidelines for fire and safety. Make sure that fire/smoke detectors are properly installed and are functioning as well as deadbolt locks on all external doors (including the garage entry door).
Check for charges that you may incur if a repair is necessary.
Check to see how much pet deposits are and if they are refundable in whole or in part.
Check to see what the charges and terms would be if you have to terminate the lease early.
Check out the community. Check out the HOA rules. Check for sex offenders or other criminals residing nearby https://records.txdps.state.tx.us/DPS_WEB/SorNew/index.aspx.
Check for the location of bus stops for city and schools and find out about schedules.
Check to make sure that the schools are the ones that you want by checking with the school district. If you are moving to a new area and don't know the schools, attending PTA meetings is a good way to find out everything that you want to know about what parents think about the schools and the community. You can also contact the school district police department and find out school safety.
Check out the Landlord by talking to previous tenants or tenants of other properties owned by the Landlord.