Hi Sarge, thanks for trying it out! You can use his gun to propel you upwards by jumping and then shooting at the ground (down+Z). Careful though, because he has to refill his ammo, so use it sparingly. Also, if you just tap the shoot button down once while over an enemy, you'll get a boost into the air!
If you still get stuck, I've put a lot of instructions on the main Spazz's Easter Caper itch.io page. Once you get the mechanics down, it's not too difficult at all :)
(I've also added a youtube video of the gameplay to the itch.io page which may hopefully help you out a bit)