Yes, your time you acquire during training is loggable time and counts toward your total flight time.
You will only log pilot-in-command during solo flight and during the checkride. After passing your checkride, you will log pilot-in-command during further training in the same category and class of aircraft as well as flights during which you act as pilot-in-command.
The minimum legal requirement under part 61 for an ASEL private pilot is 40 hours. I would shoot for efficiency and earn your certificate with the least amount of training you can, and any time you would like to pay out of pocket beyond that, fly without an instructor on board.