Yup. If hes chasing your finger around, that sounds a little more like fresh-water excitement, which is normal and perfectly fine. ;)
A good thing to look for if you're worried about stress is their color and gill movement. Unless your fish is all white or Cellophane(clear), if they're stressed they will loose color and their gills will start moving a little more rapidly, like they're taking large, quick breaths. Often their fins will clamp up as well, nice and tight together.
Also, bettas are naturally very active.....so when their stressed, they're generally be lethargic rather then swimming around a lot. Sometimes they will have bursts of quick darting movement, but often they will sit on the bottom, hide behind something, or hover near the top in a corner or something.
After a while with your betta you'll get to know his behaviors and habits and you'll learn when hes happy and healthy and when something might be a little off. ;)