Any auto parts, hardware or tool store should have a grease gun. A lot of people recommend snowmobile grease for low temps or marine grease for more water resistance. I don't think the type matters so much as long as they actually have grease in them.
I got a grease gun and multipurpose grease, when I attach the gun to the zerk fitting and begin to attempt to fill with grease it just comes out around the edges of the fitting not going into the shaft
When you say fitting do you mean where the zerk screws into the shaft or where the grease gun goes over the zerk?
If it is coming out the threads of the zerk it sounds like the zerk is loose or something is blocked up inside the shaft.
If it is coming out where the grease gun attaches you either don't have the grease gun on right or the zerk is clogged. There is a little ball inside it that acts like a valve that could be rusted closed too.
Make sure to wipe any dirt and dried grease off the zerk first too.