Shoulder/Chest Pads
I've played in pickup ice hockey games where players chose not to wear these. If you're starting out, you might consider getting a lower-end model. Since we're beginners, we're likely not checking (i.e. hitting anyone) in any of the ice hockey games that we play, so the need for protection is less. However, shoulder/chest pads protect the upper body, and it's worth the investment as you get into more scrimmages and eventually league play.
Elbow Pads
When you're holding your stick, your elbows do stick out, and you'll want these protected. Once again, if you're starting out, you probably won't need a high-end model, and I remember players not wearing these in scrimmages. As you progress, you might consider models that protect the forearms more.