Is he more of an early, mid, or late game champion or is he balanced in all 3 stages? And if he is good early how do you usually press that advantage.

he needs some levels. weak for generally first 4ish levels(?) and i think he hits critical mass at 5. not entirely sure tho, i play ADC. but iirc that's how it goes.

pressing advantage, if there is one, would be done by taking your turret (alone or with help of your jungler) and roaming. rumble is a ridiculously good team fighter with one of the longest range initiation ults in the game.

Rumble is his strongest in the mid to late game area. With a large amount of AoE damage and slows, he melts people down with just a Liandry's Torment.

And by pressing your advantage as Rumble would be to constantly zone with Flamespitter and Electro Harpoon.