I'm actually the opposite way at this point XD. Now, I actively look for some form of underlying meaning or complexity if the song doesn't already have an apparent one, and anything stubbornly simple usually gets to me in a bad way, especially lyrically. Straightforward lyrics feel like no effort was put into them, mostly because we've all heard them before at this point.
Example of simple lyrics: Ke$ha - Tik Tok
http://www.youtube.com/watch?v=iP6XpLQM2Cs
It's very predictable in its delivery, as well as "party like crazy" lyrics that are impossible to relate to unless you're an upper class 20-something year old girl who can afford to party 24-7, with absolutely no clever approach to it whatsoever. It's blunt, to the point, and overall just boring imo.
Example of complex lyrics: The Mountain Goats - Woke Up New
http://www.youtube.com/watch?v=1bSdRizGYb0
Not only is the context easy to relate to (the post-breakup empty house), he uses more symbolic scenarios to show his feelings, which are summed up by the chorus, in a way. There doesn't need to be anything complex musically, because the story being told is, in it's way, complex and emotionally gripping.
To be honest, the lyrics are probably the easiest way to start when first looking into musical complexity, because from a casual perspective, it is much easier to relate to and understand than instruments, mostly because a casual listener doesn't completely understand the variety of scales, patterns, etc that go into the instruments, so its not easy to relate to immediately.