We start with Star Trek: Enterprise (technically titled just Enterprise until season three). While the first show in the timeline, it was actually the sixth Star Trek series made, and therefore includes plenty of intentional foreshadowing of events to come.
Much of the fun of watching Enterprise first is encountering things that won't become fully significant until much later. This includes technologies in their infancy, our first encounters with races and planets we'll eventually get to know much better, and tensions between cultures that will one day become friends and allies. We'll point out some of these connections as we get to the later shows.
We'll mostly follow the original release order, but we're going to flip episodes 11 and 12 of this season, as well as save a few episodes from seasons two and four for later in the viewing order.