If there are other good primary sources I'm unaware of, it would be nice if someone would pipe up with them in the comments here. However, it looks like we only really have 2 primary sources for this movement.
The first is Josephus' Antiquities of the Jews, which makes some brief mention of a new zealous sect of Jews, which he blames the Jewish Wars on. This incidentally served to absolve the 3 traditional sects of his co-religionists from responsibility for the wars to his largely Roman audience, who were still quite ticked about the whole affair.
The second is some various debatable mentions in The New Testament. In particular John (the latest-written of the Gospels) gave "The Zealot" as a surname for one of the disciples named "Simon" (to differentiate him from the other Simon, "Simon Peter"). The others however use a different Greek word, which has also been traditionally translated to "Zealot", but sometimes to "The Canaanite" (Cana being the name of the town in Galilee where the Miracle of the Water to Wine occurred).
Much of the rest of what we think we know about the Zealots seems to have been extrapolated from the information in these sources, and other things we know about the history of the area.
The upside of this is that its pretty easy to get hold of these primary sources. There's no dearth of free New Testament and Josephus translations online. I linked to one of the latter above.
The downside is that this isn't a lot of primary sources to hang the group's rhetorical hat on. It's debatable if the Biblical sources really meant that the Apostle was part of a political group, if the John references were a mistranslation of the earlier Gospels' references, if calling someone that meant the same thing during Jesus' time as it did to Josephus, and how scrupulously historically truthful Josephus was being.
However, the mainstream view seems to remain that the Zealots existed as a movement much as Josephus describes them (what else do we have to go on, really?), and that this Apostle Simon was at one point a member.