I think you've got the basics right there, but in case you don't I'll try to explain it really quickly.
When you come down, right before you hit the trampoline, your energy is almost all KE, with only an insignificant amount of GPE. Immediately, when you hit, it acts like a spring, pushing up on you, slowing you down and building up EPE. When you get to the bottom, you have 0 GPE, 0 KE, and all EPE. Then it starts pushing on you. Additionally, you start pushing with your legs (or you were pushing the whole time, it doesn't matter.) Since you were probably scrunched up, straightening out pushes your body up and your legs into the trampoline, which acts on you with a NF, upwards. This increases the upward force, in addition to the "spring" force. Just when you get to the regular height of the trampoline, it stops pushing (theoretically speaking. The trampoline has some mass, so it would continue a little, but not much). Then, you have a tiny amount of GPE and the rest of your energy is KE. When you get to the top of your bounce, all your energy is GPE, no elastic and no KE. Then you start coming down and the cycle restarts again.
Hope that answered your question.