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
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.