is Al Ain or Kuwait a better place to teach for a single young female?

Other things being equal (in so far as your job offer and salary etc) UAE is easily the choice over Kuwait.  UAE is far more advanced, a wider spread of expat community, and a more forward-looking country.  Kuwait suffers from their nationals being pampered with lots of benefits, they virtually live a 'cost free' life from cradle to the grave.  Emaratis work much more in comparison. As for driving, Kuwait is crazy, while UAE respects road rules to a good extent, though far from perfect.  Kuwait costs for day to day stuff is pretty similar to UAE, but tends to be higher due to the inflation (caused by freebies to the locals - so expats bear the brunt of the higher costs).  Although there are no restrictions in Kuwait like Saudi Arabia, it is still more conservative.  More places to relax and enjoy during week-ends in UAE.  Yes of course you can easily visit Dubai any week-end (it is 140 km one way).

Ah-one more thing.  In UAE (esp Dubai) it is far more easy for a single woman to be moving around on her own.  Not so easy in Kuwait.  The local lads are a nuisance on the roads in Kuwait.

Thanks so much for ur help! My potential salary in al ain would be 8500. Employer is paying rent and all utilities. I just have to get  car. The place is furnished.

Without doubt I would choose Al Ain. It's a clean city with a lively expat community. Kuwait I found to be dirty and very restricted. 8500Dhs. is good enough to survive and either have a hire car when required or a monthly rental for a 1.4 car would be no more than 1,500 a month.
Fee

Thanks sushi fee and everyone else! Do you think u would be able to save a good bit each month from the 8500?

that's the eternal question that only you can answer nadia! Depends on how much shopping, partying and sightseeing you plan to do while you're here. If your intention is to save, then put aside a set amount every month not to touch and spend the remainder

If you live smart, you can save 25%