From my understanding all I need to do is tie up some shrimp (what kind of shrimp should I use? From a local fish market?)
Yes just ordinary shrimp that you buy from the stores or markets. You could even use a piece of fish. I used mussle meat when I did my cycle. One would be enough for the cycle. Also, to help the cycle you could use old filter media, decorations and gravel from an old already estabilished tank, if you have one.
In a pantyhose and let it sit there until my ammonia and nitrite are 0 and my nitrate is below 20ppm for several consecutive days.
You don't have to use pantyhose its just easier to take the shrimp out of the tank and won't make a huge mess.
I know I need to change the water while I'm doing this but I'm not sure how often or how much water to change.
When cycling you don't need to change the water, just let the tank cycle. When it's complete you can then change about 30% of the water.
Should I use the Seachem Prime every time I make a water change while cycling my tank?
Unless you have fish or plants in the tank when cycling, then you don't need to use this just yet.
Also, what kind of filter media should I use? I was thinking carbon. How often should I change the carbon? What carbon would you recommend?
Most filters that I have used come with the carbon already. I'm not too sure about this stuff, I personally don't like it but I'm sure someone else can help you with this. When the carbon runs out, it will turn your water a slight yellow, this is when you need to change it.
I want to put some live plants into my tank. I know nothing about this so any info would be great. Will putting live plants into my tank effect the cycling process?
Adding plants when cycling will help the process by adding bacteria, if you do make sure the plants are hardy like anubias or java fern because I'm not sure if the cycle will damage the plants. Again I'm sure someone else can help you
The 25W heater will be perfect for your size tank. I use a 25w heater in my 8 US gallon tank and it stays a consistant 80F