Wandering around and fighting random thugs has begun to grow a bit repetitive, and I can't find any doors that I can enter in order to explore new areas. I found a couple of places where it looks like I can use something to break the wood, but I'm not sure how to do this. Should I be approaching this more like a linear game, or am I just not finding the right buildings that can be entered?
If you follow the main storyline it will take you across parts of the city that will open side quests, which in turn can lead to finding more side quests. Just continue the story and bounce across roof tops and use the detective mode to find accessible doors (big, orange, hard to miss)
The wood needs to be broken with the explosive gel sprayed directly on it. Another method I founf recently is the Zip(line) Gun if you aim and fire at the wall you will zipline through it all heroic like

EDIT: There are Riddler Challenge rooms which you need to break the wall to enter, these wall/doors just need to be "activated" in order to break them down, they don't show up blue with a crack though but rather with a Riddler question mark on them.