Yep, you have 8 sprockets. They might have a number stamped on each cog if you clean it up and look closely. This will be the no of teeth.

You have a triple at the front which should mean you have some nice hill gears, depending what your largest ring at the back is. Doesnt sound like you need to change anything necessarily, so it could just be a case of getting over the mental aspect and maybe working on those steep sections with adifferent technique - do you predominantly stand or sit or mix it up?