Trust is very relative...
MtGox has 85% of the trading volume so if you want to trade frequently, that might be a point to consider. They got hacked a month ago and they recovered, but I don't know if the money was fully restored to users. There are also some complaints about their customer support...
Tradehill is the second largest. I used it a couple of times and so far so good. They talk a lot here at the forums so it's easy to know what's going on.
CampBX is recent and it's building up volume. They seem to be a serious and professional company with a great focus on security. They also plan to add advanced trading options, such as short selling, as soon as they get enough volume. Good customer support.
There are more, but these are the ones I tried. As for getting dollars in/out I can't help you with that (I am European). Apparently most people use Dwolla because it only costs $0.25 per transaction, but Tradehill no longer accepts them.
Most secure place to store bitcoins: in your PC disconnected from the Internet and locked up in a safe.
I have most of my bitcoins at the exchanges because I trade frequently. I guess you have to trust them a little...
