Run the battery you have and see how it works. Yes a deep cycle battery is better, but the battery you have is already in the trailer so you might as well see how it works for your style camping before you go spend money on a new battery.
To hook the battery up there should be a few wires near it. It's extremely important to make sure the wires go on the proper posts on your battery.
As Old-Biscuit said you need to identify your negative battery cable. If there are two wires near the battery grab the white one with your hand and follow it and see if it is connected to the frame. Then follow the black wire and make sure it doesn't also go to the frame (it most likely does not but it's good to check.)
If the white wire went to the frame it is (negative or the green side of the battery in the photo. If the black wire does NOT go to the frame it is positive or the red side of the battery.
You might need a some battery terminals to allow you to connect the wires. These are available at Walmart or any auto supply store. Make sure you get the ones with the wingnut on them as it makes connecting and disconnecting easier.
