Here is the info from the state's website on the process of getting a HCP - Handgun Carry Permit.
Basically you have to take a state approved Handgun Safety Course...I know of two ranges in Memphis that offer them.
Take that certificate and $115 to a full service DL station, fill out the application, get finger printed, turn all that in and wait up to 90 days.
As far as recommending a pistol....Not something I usually do because everyone has different taste, I carry a Glock 32 (Compact .357 sig) I think they are around $475 new.
J-Frame S&W revolvers are nice too if you want a wheel gun.