The proportionally strongest guy in my class at PT school does only martial arts bodyweight work, so that alone can make you pretty strong.
However, if that's not enough... A 5x5 programme is excellent for developing strength. On this forum, many of us follow that programme or something similar. This will also improve core strength. If (for example) a wall squat with a fitball and zero kg builds core strength, what does a squat with 80kg on your back do?
For truly explosive strength, or "muscular power" as it's fancily called, I know that weightlifters (ie "olympic" style) have this in large amounts, but are not always very big in size. On this forum, we have at least one weightlifter, Nick (NPR), I'm sure he can comment on that more usefully than me.
The key thing for whether you get bigger is diet. Your body must have raw materials to build more muscles. If it doesn't, you may get stronger but won't get bigger. There is of course a limit to this, strength and size go together - that's why your sport and weightlifting both have weight divisions - they're just not perfectly together, if you know what I mean.