It's very low amperage, so you can use 14 or 16 gauge with no prob. Just don't touch the front terminal (where the grey wire is) to the back terminal (can't be seen in your photo so I don't know what color it is) with a small wire. It will melt in your hand. You don't have to replace all of those today, but do make sure they are clean with a wire brush or something.
I know I wrote alot, but I was trying to give you a step by step with most possible problems so you won't have to keep going out there and then coming back in to see the next step.
My guess as to the problem, in most likely order is,
bad battery and poor/dirty connections
or
bad starter
or
bad solenoid
Don't just go changing parts, starter or solenoid, if you aren't sure that they are bad. I reccommended changing the battery cables, and the terminal lugs as a reliability thing, and it is real easy and cheap. I see alot of probs. in the shop from corroded/poor connections.
Good luck!