Is there an underlying story arc in Supernatural?

You're actually pretty spot on with your assessment of the beginning of the series. However, a look at Wikipedia's Supernatural entry gives us some insight:

According to creator Eric Kripke, the show originally was intended to focus on the weekly monsters, with Sam and Dean Winchester merely being "an engine to get us in and out of different horror movies every week". His sole desire was to merely "scare the crap out of people". However, a few episodes in, Kripke and executive producer Bob Singer noticed the onscreen chemistry between Jared Padalecki and Jensen Ackles. This revelation caused them to change the series to focus more on the brothers than the monsters, basing the weekly monster around the storyline they wanted for the Winchesters. According to Kripke, "...sometimes we don't even have the monster until way late in the break, once we get all the angst and the drama done first."

Unlike shows with "endless mythology" like Lost, Kripke prefers to keep Supernatural's mythology simpler, saying, "It's so hard to go season after season after season with a mystery and then provide an answer that's going to be satisfying." He prefers to have the series' structure like that of the earlier X-Files episodes, having mythology-based episodes spread through many self-enclosed episodes—Supernatural usually having three self-enclosed episodes followed by a mythology episode. With this format, viewers do not have to have previous knowledge of the mythology in order to watch the series, being able to "join the party at any time".

So, to summarize, the series did start as a monster-of-the-week show, but developed into a show about the brothers. They kept the format of 3 monster shows - 1 "canon" show; however, they started including a lot of story line around the brothers later into the first season.

Keep watching, it only gets better.

Short answer: Yes, there is a continous plot through main characters.

Longer answer. It's right that at the beginning there was plenty of just random single standing episodes (as stated by DForck42l quoting the makers of the show). And this has happened through out the series quite a lot. But when the main plot comes forward there's very little room left for single monster hunt episodes with no background.

However, even at the beginning there were always more or less a certain amount of episodes that came to be the main story of the series:

How the brothers can save their father (seasons 1-2), each other (always), themselves (few times when they take separate ways, or dream of doing so), their extended family (their step brother and close friends), and ultimately all human family (when apocalypsis and darkness) and perhaps even all kind of spirits (taking care of Castiel, the angel; and even trying to bring the best from the most evil, like when they tried to restore the evil Crowley's humanity).

Summing up, the main-to-filler episodes ratio varies according to the season. Seasons 3-5 have an increasing amount of continuous plot story and the strongest overall. In some seasons this gets a little worse. But all in all, the answer is yes, there is at least one underlying story arc, and perhaps several subplots always going around different characters, subjects, and mythologies in general. But monster-of-the-week is not the idea of the show once you get past the first two seasons.