The manga is the same on this point: the characters call each other by given name pretty much from the start. I think there are a few reasons, in-universe and out-of-universe, why this makes sense.
Mio and Ritsu have known each other since they were young, as we saw in an extra story in Volume 3, so it makes sense that they would already call each other by given name. Given Ritsu's personality, it makes sense that she would immediately jump to calling Yui and Mugi by given name, and neither of them are the type to object; Yui doesn't stand on formalities, and Mugi is always looking for ways to get closer to her friends. Azusa is an underclassman, so she would be getting the more familiar forms of address anyway.
Another thing to keep in mind is that time passes in strange ways in both the anime and the manga (but especially in the anime). Azusa appears in Volume 2 of the manga and Episode 8 of the anime, so an entire year passes in one volume / seven episodes. In the anime, Episode 4 covers the summer training camp at Mugi's villa; since the Japanese school year starts in early April, the girls have already known each other for at least two months by this point. Episode 7 is Christmas, so by now they've known each other for around eight months. If I were the writer in this situation, I wouldn't think it was worth it to have them all call each other by family name for five episodes and then suddenly have them switch to given names; it would be annoying and confuse the audience, who may not have taken the time to memorize everyone's given and family name right off the bat. (For instance, I've read the Genshiken original manga series about fifty times, but I couldn't tell you what Kugayama's, Tanaka's, or Kuchiki's given names are without looking them up.) Same thing if I were writing the manga; I would not have thought it was worth it to switch everyone's form of address halfway through Volume 1 when they've known each other for six months.
It's pretty common in "cute girls doing cute things" series for the main girls to all call each other by given name immediately upon meeting and forever thereafter. Everyone in Hidamari Sketch does it, even though Hiro and Sae are upperclassmen. (The Hidamari girls do use honorifics, though, unlike the K-On girls.) Same for Is the Order a Rabbit?, Kiniro Mosaic, Lucky Star, and Yuru Yuri. The only exception I can think of is G.A.: Geijutsuka Art Design Class, where everyone refers to Tomokane by her family name. My theory is that it fits the atmosphere these shows are trying to create. Part of the appeal to this type of series is the sense of being among a tight-knit group of friends, the kind of group where everyone speaks informally. It also makes the characters seem guileless and child-like; like children, they're living in a little bubble where social standing and deference aren't so important.
In more realistic anime, like Honey and Clover, or workplace-oriented shows like Wagnaria!!, characters call each other by family name as they probably would in real life. But for a show like K-On, realism is less important than creating an atmosphere, one which is harmed if the characters are too distant with each other. And the characters' personalities and prior relationships permit that sort of break from reality.