Safe inexpensive 4 door for mother of 1, under 6K, reliability and what we should look for

For this there is only one answer. The Toyota Corolla. It is the appliance of cars. It is very safe, it is very economical, it is one of the most reliable cars on the road in the USA, the AC is fine, and it is not that expensive. You can get a 2007 to 2010 with 70,000 to 100,000 miles on it in your price range in your area, and those cars can go over 200,000 on the original powertrain easily.

Don't expect a lot of power but enough to merge with fast moving traffic. Don't expect a lot of space but enough to be comfortable. Don't expect luxury as it is basic transportation. But as a first car it is hard to beat.

Back in the 1990s I owned one as my first car. The thing was indestructible and it fits exactly what you describe. Today they are just as reliable and durable.

There are other cars like the Mazda 3 and Mazda Civic as well if you prefer them which are nearly as reliable as well, I just think the Corolla is the best choice of the three.

Best of luck.

AcuraT's recommendations are excellent. I would add to his list the Mazda 6, Honda Accord & Fit, Kia Optima & Soul, and Hyundia Elantra. It is surprising how many pretty good cars are available on the Seattle Craigslist for $6.5k or less. The key word here is "good" because they are not all good.
How do you find a good one?
You need to know or have someone who knows cars assist you and drive the car for at least 10 miles, test all accessories. You can google "how to pick a used car" for more info. Consider using Consumer Reports and True Delta to find the most reliable years and potential trouble spots. If you find a car you like but have doubts, spend $75 for a mechanic to check it.
I'd go to craigslist, type $6,500 in the price space, then type in the model -- "Corolla" for example cars & click the icon near the top that says "owner". You want to look for an original or 2nd owner car (who has owned it for at least 1 year - not a flipper). You want someone who has all the service records & has changed the oil at least twice a year. You can't trust all original owners, but many of them have loved their car, take good car of it & just want it to go to a "good new home".
IF it has over 90,000 miles find out what has been replaced. Most cars by that point need a timing belt, spark plugs, brakes, AC service, coolant changed, & sometimes CV joints. IF those haven't been replaced within the last 30K, take it to a mechanic for inspection and expect to spend ~ $500 to get certain parts replaced so the car will be reliable.
I'd be looking at the above models starting at $6,500 as most owners ask high expecting to get a little less.
I wish you well and would love to hear what you end up with and the process.