New to Arduino - Looking for advice

I'm in Canada, I will do an internet search for a supplier here.

I will likely only be using the board for this and future generations of this idea. I assume I can reuse the board again? Something small is preferable as space is limited inside the structure. I was thinking of using a 9V battery to start and then refining the power supply to something smaller and likely more suited to the application. If I unplug the battery from the board, will it remember the program? Sorry this is very new to me; I usually deal with biological research.