Which Aduino do I need?
There are many, many different types of Arduino boards, but the most-used, officially-support board is the Uno: https://www.adafruit.com/products/50
What else do I need?
Without knowing more about your project, I wouldn't feel too comfortable giving you a list of anything to buy at this point. I would highly recommend figuring out more details about your project before thinking about hardware or things to buy. For example, how do you plan to interface with your cameras? Is there a standard plug or connection, or are you hacking your cameras in some way? Do you know how digital camera timers work, or if there already exists a programmable timer that you can just buy for this purpose?
What do I need to know? / Where will I learn that?
Based on your last paragraph, my intuition tells me you could use an LCD and a couple of buttons to increase or decrease the 'time-parameters' visually. Then, you can connect some switching circuitry (depends on how much current you need to switch) to your cameras' interfaces. Not being a photographer, I don't know what 'signal trigger inputs' are, but you can easily take analog or digital measurements with the Arduino and utilize those values in your code.
Can you explain your project a little more, and maybe provide some relevant examples so we know what we're talking about? Also, could you tell us a little bit about your experience level, and how much electronics experience you have?