As per getting experience or starting right away, it comes down to what works best for you. I worked as a Basic for only a few shifts before I started my medic program, and did fine. I know people who were EMTs for years, and did fine, and I know people who took their EMT 4 years ago, never worked in EMS, and did fine. There is no "correct' way of doing it, aside from starting when you feel comfortable.
As far as schools go, since you're in Cleburne:
There's Hill College, it's the closest one to you. I have no experience with their program.
There's TCC, up in Ft Worth, which is where I got my EMT from. They are renowned for their paramedic program as well. They do 2 classes a year, one in January and one in August, the January class is the smaller one.
I got my medic from EMTS, but that's over in Dallas which was almost an hour drive for me, so it'd be a bit further for you, but I would recommend it if you don't mind the drive.
As far as online goes, there's trainingdivision.com. They are actually based in Crowley, so it's online and it's close to you. I've heard decent about them.