With more riding will come confidence in riding at speed. Until then, just go slower.
Old bike is OK, but bike that does not fit is not. I sounds like your bike is way too big! You will be a lot more confident on a bike that fits.
Regular pedals are fine. You can try the special "clipless" pedals and shoes once you get more cycling experience. Or not... I used them for mountain biking, but for the casual, non-competitive cycling I do flat pedals are just fine.