To be clear because the title is slightly ambiguous: I understand that to check for collision detection it's (box.x > otherBox.x && box.x < otherBox.x + otherBox.w) etc.
However I'm curious about things to do after this, how can I sort of keep the player from entering the other object that they're supposedly colliding with without fully stopping them and therefore suspending them from movement?
You would only restrict movement in that direction. In other words, let the player move left only he has not collided with a wall to his left. Let him move right only if there is no wall to his right. This way, if there is a wall on one side, the player can still back away.
You would do this by checking multiple collision points, so instead of box.x, you would check both box.x+(box.width/2) and box.x-(box.width/2).
