Congrats on starting running and your post cracked me up!
I saw an article recently on how to find a good bra, but I can't find it at the moment. If/when I do I'll post the link for you to read yourself, but the main points I took away from it:
- You're looking for a bra for "full support" or "high impact," whatever the highest rating is.
- stretch the material of the cups up/down and left/right. It shouldn't give very much at all. Same with the straps. Stretchy = NOT good.
- you should be able to get two fingers under the straps but it shouldn't be loose.
- Cross-back straps are probably the best support option (as opposed to the U shape like a regular bra)
- Make sure you get a good cup size - too big and you'll bounce all over; too small and it squishes

Compression is good!
- Don't wear the same bra twice without washing. I did that a few times and regretted it
- Wash: cold water, regular detergent, but do NOT put it in the dryer. That will kill whatever support pretty quickly. Just hang to dry.
I found a great bra at Target on the sale rack for $8, just testing them out. I doubt that will happen again for a while

but there were several kinds and not all of them were the high impact, but you
can find them for less. I second the many other recommendations for Moving Comfort as well.
ETA: ok here are a few articles that repeat some of the other things I read:
http://womensrunning.competitor.com/2013/06/shoes-gear/gear-girlfriends-guide-to-sports-bras_13158/3
http://www.fitnessmagazine.com/workout/gear/sports-bras/how-to-choose-a-sports-bra/
http://www.rei.com/learn/expert-advice/sports-bras.html