I'm watching the third season of the original Star Trek...should I have started with the first?

I started watching the series on TV every Saturday night at 11:00 PM and fell in love. (I would definitely call it one of the highlights of my week.) However, I began in the middle of the third season and am almost certain that there not being show in the correct sequence. I enjoy the anticipation of every new episode and staying up a little too late to watch it. But if I am not getting the full Star Trek experience by not watching them in order, then I will star watching the episodes online.