The grain in film is caused by the silver halide crystals (the part that changes when exposed to light) in the film's emulsion. You probably used a high-speed film. High-speed film is more sensitive to light, due to the large crystals, resulting in grain. Film with an ASA/ISO rating of 50-125 will have very small crystals, resulting in negligible grain. They are, however, less sensitive to light. Use a film that is optimized for your lighting source (natural light, incandescent, etc.), so that the colors are rendered faithfully. Light your subject so that it looks good to your eyes, then shoot without a flash, preferably with a tripod. If you have no tripod, place the camera on a solid object so it doesn't move when you depress he shutter.
When using the zoom, I would suggest zooming all the way in, then moving the camera to frame your subject. Most zoom cameras, when zoomed all the way out, are very wide angle, which makes near objects large, and far objects small. This can be seen as "bending" around the frame of the picture.
For film use, a "gray card" (Ansel Adam's Zone 5), in an indespensible tool for getting an accurate light meter reading. They are available at better camera outlets. I NEVER shoot 35mm film without one.
I've had 3 semesters of Photo in college (A, B+, A), and I still learn every time I take a photo.
Good luck with the pictures !