I want to get into rental properties, but how am I supposed to make money if I end up with these dead-beats that don't pay rent and only destroy the house/duplex/whatever?
Get good tenants.
How often do you realistically encounter these people and what kinds of problems have you had to deal with.
Very rarely, if you screen well.
\Also, what are some stories of GOOD renters that you've had and how did you find those people?
They're boring stories, which is why you don't hear them. My tenants paid me on time, rarely bothered me (only for major things, fixed minor things themselves), then left the house in great shape when they moved out.
...
/shrug
(True story, btw. Many times.)
Is there a common trait to look for in order to find a good renter?
Yes, tenant screening is both a science and an art. And there are laws around it (fair housing laws). Do lots of reading on the topic, because tenant screening is one of THE key items to being a successful landlord.
What do you recommend as far as purchasing property? Should I go to the really cheap places to start or buy into a nice area that's a lot more expensive and less likely to get the rent I am asking for?
I would recommend educating yourself on rentals before doing either of these. You'll need to define your goals before making a decision like this, but if you're at the stage of asking this question, you're at the stage where you need to learn more so you can define your goals.
Should I even bother with getting a rental property? I am not handy AT ALL. I would have to contract out EVERYTHING if the place got destroyed, which would cut into my profits. Should I just buy more REITs or stocks instead?
I don't do any work on my rentals. I'm not handy at all either. Heck, I don't even pick up a paintbrush, though I'm pretty sure I could do at least a mediocre job at that.
Build it into your numbers. (Which you should do even if you are doing it yourself - because at some point you may have to or want to pay someone.)