Hi Heather, this tutorial has loads of information you can use to clean your carpets without a carpet cleaner- https://www.firstforwomen.com/posts/how-to-deep-clean-carpet-163019
You can rent a carpet cleaner at most home improvement stores like Menards, Home Depot, etc. WalMart has them as well and so do many hardware stores. Renting a steam cleaner is the best way to get your carpet really really clean and it is cheaper than hiring a professional carpet cleaning company. Or perhaps you know someone who has a carpet cleaner and you could borrow it.
You can purchase a good spray cleaner that vacuums up. IF you have spots, SPOT SHOT is really good for removing those. You can also rent a cleaner. I purchased one for under $100 at Ollie's and use vinegar to clean mine regularly.
For a quick albeit temporary fix, sprinkle with baking soda, let sit over night and vacuum up in the morning.
Have your carpets professionally steam cleaned once a year for optimum performance. I have mine done every 3 months, but I have a lot of pets.
I wrote a post about how to spot treat with 2 ingredients and you can clean your whole carpet this way too. https://chascrazycreations.com/carpet-spot-remover-with-2-ingredients/
I have used the baking soda technique at times or a flowery (basically a scented baking soda product from dollar stores), sprinkle on and vacuum up. It has worked well.
My mother taught me years ago to clean a carpet in the snow-assuming you get snow where you live. Take the carpet outside and let it lay on the clean snow, then beat the carpet with the flat side of a broom to knock out any dirt or stress you have. Turn it over and repeat. Then cover the carpet with up to 6" of clean snow. Let it set covered for at least 1/2 hour, take the broom and remove the snow by brushing it off. Bring it back into the house and put it down where it came from. It will dry on the floor.
It will smell so clean and fresh.