I assumed that you were already a rated pilot... (I guess I shouldn't do that) You can still take lessons and can still log your time as dual received. You can't log PIC time since you aren't rated. But you won't be able to solo without that medical certificate. (my opinion here, if you're already denied, I am assuming again that you haven't spent much money towards flying, so you should clear up your medical situation before you spend any more money on flying...)
There is a sport pilot certificate (e.g. the driver license medical - if you have a driver license, then you can fly with that) that just became effective, but you would be ineligble for the sport pilot certificate as you've been denied a medical certficate. You can still fly gliders without a medical certificate.
AOPA will not get you your medical... they don't do any medical tests or anything. You still need to go to an AME, but AOPA can give you advice about your situation that is probably more relevant to you than what I or anyone here can give you.
As far as looking for an AME, you can ask around locally or you can look for one at http://ame.cami.jccbi.gov/
If you can't find any info as to which ones were good, you could call each one and ask if they would work with you to clear your situation.
From the AME's I've visited, it appears most AMEs do it on the side as well as their normal medical practice, it's just that they've gone the extra step and gotten FAA approval to do flight physicals.
Woodreau / KMVL
Bonus animus sit, ab experientia. Quod salvatum fuerit de malis usu venit judicium.