bike going wobbly? sounds like shimmying (sp). Typically happens at high speed and on corners. To stop the bike from shimmying, stop pedalling and press both legs onto the top tube of the bike. The legs act as a damper and will smooth out the shimmy.

Make sure your head set is on tight, see that the wheels are true..

As someone said, put your weight toward the back when you have to hit the brake. This will keep you from getting the feeling of doing the endo (end over -- flipping over the handle bar).

Practice riding behind an experience rider to develop your confidence.