Most people say that Burma is generally very safe from violent crime in the main cities, although as the tourism industry builds, there have been some incidence of scams targeting foreigners, including a few (but not all) money changers, etc. Just keep your wits about you, and as a foreigner, I would be careful to keep my documents in a safe place.
During my week in Myanmar I saw several local people, men and women, carrying wads of cash in their longyi's or in bags, even around the train station, so they didn't seem to have fears of theft (although you should always be careful about pick-pockets, no matter what country). I would be careful to avoid political discussions in public not so much for your own safety, but that it could unwittingly get the local people you spoke to in trouble. Outside of the big cities, precautions against mosquitos at night should be taken because I have heard there are some malarial areas.
In Bagan and other tourist destinations the guesthouses tend to work with other businesses and try to direct you to their partners even when that isn't serving your interests of time, money, or independence (not a crime, but something of which to be aware anyway).
I'm not sure if it would be helpful to you, but Guy Delisle wrote a comic book (graphic novel) about his experience spending a year in Myanmar -his wife was with Doctors Without Borders, and he was raising his toddler son. It is published by Drawn and Quarterly books. It had some interesting cultural info.
