I went to Prague last year with my husband and went out a couple of times on my own and I have to say that I felt perfectly safe. Infact someone stopped me for directions and I had to say that I wasn't a local so I think I'll take that as a compliment
This site here will give you details of sightseeing tours as that would be a good way to meet people and share experiences -
http://www.pragueexperience.com/sightseeing/sightseeing_tours.asp
Obviously as in any major town petty crime can be a problem but just take the usual precautions and you'll be fine - wear your bag crossways, don't carry much cash etc. I'd stess that we never witnessed any crime.
If you are on your own I'd book a taxi from the airport to your hotel in advance, that way you are not messing about once you arrive. We arranged ours via our hotel and the driver met us in arrivals.
If you visit the astronomical clock, I'd be interested to hear what you thought. We thought it was a bit of a let down and noticed that lots of people around us had expressions on their faces as if to say "was that it??"
Have a good time.