First of all, get a good all-purpose cleaner. I'd recommend Mothers. It is a gallon size and would need to be diluted with distilled water (probably around 7:1 or so for your purpose). Another choice is Meguiars, the one I've used for 2 decades. However, Meguiars has a strong fragrance odor that has caused me to switch to Mothers.
If you don't want to go the gallon/dilution way, Home Depot has Spray 9 and Odobon -- both ready to use cleaners. I like Simple Green for most things, but not this job.
Second, get a good brush. For this job, if you're going to do it by hand, get the Oxo brush below:

Its heavy enough to get the job done. Somewhere between a carpet and upholstery brush, but for your purposes, it will work well. If you want to use power, Amazon has brushes you can attach to the end of your drill. I use a brush on my Porter Cable polisher. It is a little less stiff than the Amazon drill brushes and I can control the RPMs a little better on the PC. I got it from a detail shop online -- make sure it is the one that will attach to a RO polisher.
Third, use the cleaner and whatever brush you decide and go to town. Scrub like there's no tomorrow. Get the cleaner into the fabric but don't over-saturate the fabric with soap/suds. The thing to keep in mind here is the fabric is tough. You will not tear or damage it with rigid scrubbing.
Finally, use the carpet extractor tool you mention and suck all that stuff up. Go over it multiple times until all you see is clear water coming up. Note that your seats WILL be wet and you'll need to let them dry. You also might want to run a steamer on them to neutralize any bacteria but the cleaner will take care of most of that.
The whole project will take you 2-ish hours to do it right. Go ahead and do the carpets while you're at it.