I'm fairly new to cycling, too, and was pretty timid on the roads when I started out! One of the things that has given me a lot more confidence on roads was to ride with a group of people. I joined one of the local cycling clubs and ride with them often, but I find that even cycling with just one other person makes me a lot less nervous about whether or not drivers can see me.

On steep downhills, one of the things I've heard that's helpful is to slide back a little on your saddle. Transferring your weight farther to the rear of the bike makes it more unlikely that you'll go over your handlebars.