If you're going to do track days before dipping your toe in the racing pool at least get yourself a good laptimer. IMO if your goal is to race, then you don't want to
feel fast, you actually want to know if you
are fast. A laptimer is the no BS way to objectively assess your performance. The more sophisticated laptimers will give you the capability to assess your speed at any given point whereas a simple laptime will only tell your overall performance. Certainly at first, just knowing your laptime will be good enough to let you know how you are doing without overloading you with too much information. You can leave the corner by corner analysis for later.
I will say that there is something to be said for simply taking a race school and starting racing right away. It kind of forces you to figure out how to improve in order to be more competitive. That is unless you like getting beat!