Oh, you can go out by yourself, it's safe enough.. just be prepared for some oddities. People (men, mostly) may come sit with you at restaurants, and then expect you to pay for their meal. As a Western person, you have the highest status and perceived wealth. If you don't mind them tagging along, they may also expect you to pay for bus fare, etc. Don't worry, it's all so cheap.. you can afford a few hangers on.
Any visitor will have a few problems with the taxis.. they're all going to want to stop at silk shops/jewel shops on the way. If you're really in a hurry, demand to go straight to your destination. If you're not in a hurry.. you can negotiate discounts for stopping, the drivers get vouchers for delivering the tourists. The shops are expensive, but you might like to go, anyways.. you don't have tons of time for price-shopping, and the selection is amazing.
Dangers for the solo female.. not so many, really. Avoid gambling parties, watch your drink, the usual. Leave any place that has obvious drug-dealing going on. You can safely eat and drink stuff at the street stalls. A lot of bars have semi-naked pole dancers, but it's no problem to go in for a drink. You'll have to pay for your hostess the same as a guy would, but it's all fun.
I love Thailand..
Oh, temples. You'll find a lot of 'guides' near the entrance. They're not a bad investment, really, but I wouldn't go to a fortune-teller there. They inevitably tell you to release birds or live eels at some point in the rather distant future.. when you're back in a country that doesn't have live eels for sale..