It's very normal to feel nervous about showing. The rider/handler is putting him/herself under the microscope, performing the culmination of months (more frequently years) of work out there to be judged. Most people I know (myself included) are not only exhibitors, but the owners and trainers of their mounts, hence being judged on more than just their riding ability, in a way, as they are responsible for the condition and behavior of the horse.
Preparation is the key to a relaxing and positive show experience. Check out the rules/expectation for the classes you're riding in. If you're signed up for hunter over fences, but you and your horse are really new to jumping courses, maybe try hunter hack instead, or just a flat class. I'm not saying show below your level, but don't go into a class for something that you can't nail at home with relative ease. In Dressage, I believe the saying is that one shows a level below what they're schooling. Be at the show grounds at least an hour before your first class; this gives you plenty of time to sign in and get your horse settled and warmed up.
For me, it's the anticipation that makes me nervous. Once I'm in the ring, my brain clicks and the butterflies settle. I usually can't eat on show day without feeling a touch queasy, although I try to get something down if I can for the sake of not getting sick that way.
The thing that helps me if I get really nervous is to remember that "It Just Doesn't Matter! It Just Doesn't Matter! It Just Doesn't ...".
At the end of the day, you're not going to be any worse off than you were at the beginning. Just relax and enjoy the opportunity to show off your horse and what the two of you know can do together.
Hope that you're able to lease a horse and get in the ring and experience it for yourself. Best of luck!