First, you'll need to have a little chat with your local extension office to find out what kind of turf is best suited to your location. They will probably encourage you to take soil tests.

Then you can tackle the problem of the weeds. Whether you find some sort of variety of grass to seed OR to lay sod, you will need to get rid of the weeds and do some prep. work first. Your extension office should be able to provide you with plenty of information regarding the establishment of warm season lawns, as well as weed control.

Ornamental turf species can be divided up into two main categories: warm season grasses and cool season grasses. Your options consist only of those species that can be termed warm season varieties. The better of those are all vegetatively produced, not seeded. That means sod, rather than seed.

You will probably narrow your options down to Centipede grass, St. Augustine, or one of the Bermuda hybrids. You CAN opt for Common Bermuda, but be prepared for almost constant seed heads.

By the way, a twice a month mowing schedule isn't a healthy one for actively growing turf and it sure won't help keep the weeds from going to seed.

Here is a link that might be useful: Find your local extension office here