I'm writing this after shampooing the carpets and cloth-upholstery on my wife's 1996 Cavalier convertible, but it should be applicable to many similar car models and years.

My wife's carpet and cloth-upholstery were disgustingly dirty and stained from 18 years of too many food & drink spills, too many dirty (and lotion-covered) hands touching the upholstery, and too many people putting their dirty, greasy shoes on the carpet instead of the mats. I was going to take it to a detail shop but I decided to try cleaning it myself based on some DIY tips on the Internet. It made a HUGE improvement. It's not perfect but it's perfectly acceptable. I don't want to rehash the shampooing instructions in detail because you can look them up on YouTube, etc., but briefly you:
1) Vacuum the car with a good vacuum, like a shop vacuum (in dry vacuum mode).
2) Spray on shampoo/carpet/upholstery cleaner.
3) Scrub the carpet and/or upholstery with an upholstery brush.
4) Vacuum out the shampoo with a wet/dry vacuum in wet mode or a steam-clean vacuum. Optionally you spray it with water and vacuum it out a few more times to remove any shampoo residue.
5) Let it dry.

However, I did want to offer a few "non-standard" tips that may help those who want to tackle this job...

* I think it's much easier to shampoo the carpets and to clean the seats by removing the seats. I was shocked to find that I could VERY EASILY remove the front seats on her 2-door convertible were merely removing 4 nuts. (The bolts protrude from the floor and are part of the car's body.) Removing the seats gave me easy access to the entire carpet, and it let me easily clean the sides of the center console. It also let me clamp the seats to a workbench and clean them without having to bend over. (Note that the driver's side seat had an electrical connector for the seat belt sensor, which is very easy to remove.)

* When shampooing, I used the "Chemical Guys Carpet Brush for with drill attachment" mounted on a cordless drill. It was $10 on Amazon. Worth every penny. I think this made the job much easier. (I'll probably start using it to scrub my shower too.)

* I also used Chemical Guys' "Fabric Clean" cleaner for the carpets and upholstery (based on the positive Amazon reviews). It worked well but I don't know how it stacks up to other carpet/upholstery cleaners.

* You may need to shampoo stubborn stains more than once, but some old, ground-in stains won't come out, and I found that after the second shampoo, it's probably not going to get better doing it a third time.

* It's probably best to do this job when the weather is nice and you can give it a full day (or two) to air-out and dry.