I'm in East Ft. Bend (Katy).

I had great luck with Sunquat--the tree is constantly covered with little limes and started producing fruit when it was only knee-high.
Mexican (Key) limes are also supposed to do well around here.
Satsuma mandarin oranges do great, too.
Citrus need to be protected (blankets) from hard freezes, but if the tree is too big for a blanket it's usually OK.

Avocados grow around here, but don't usually produce fruit.
There was an article in Texas Gardener magazine a year or two ago about a guy trying to raise avocados, but I don't know how it worked out.

Figs, persimmons, loquats, and pears are also really easy (no spray, no fuss, stick 'em in the ground and wait for fruit.)

The above are very good suggestions. You also might want to give the Fort Bend County Master Gardeners a call and ask for some suggestions.

January & February are 'fruit tree sales' season around here. There are several 'sales' this time of year. There are several avocado, lime, and lemon trees that will do well in this area. These tree sales only sell trees that do well in this area. I went to the Urban Harvest sale a couple of weeks ago and they had several varieties of avocado trees. Do some research and find one of these sales. They are in galveston, brazoria county, montgomery county and others. Good luck.

There are a couple of great nurseries up by you with varieties of both lime and avocado that will do great for you. The rare fruit growers meet not too far from you also. Check the Tx garden formum-Ed posts the days there, maybe on the exchange page.
JnR and Caldwell nurseries. You can google them both.
Tally HO!

Mexican Type Avocados are suppose to produce in about 2 years and are a hardy tree. I understand they only grow to about 8' and aren't not good for commerical use as they are too perishable but great for home use. They are sold at the Fort Bend county sale every years but they are suppose to be available in many nurseries in this area.