We study the simultaneous actuator selection and controller design problem for linear quadratic regulation with unknown system model. We propose online algorithms to solve the problem by interacting with the system in an online manner, and the algorithms specify both the sets of actuators to be utilized under a cardinality constraint and the controls corresponding to the sets of selected actuators. We consider both episodic and non-episodic settings of the problem, where the interaction with the system breaks into subsequences in the episodic setting, and the interaction goes on continuously in the non-episodic setting. Our online algorithms leverage a multiarmed bandit algorithm to select the sets of actuators and leverage a certainty equivalence approach to design the corresponding controls. We show that our online algorithms yield sublinear regrets with respect to the horizon length considered in the problem. We also extend our algorithm design and analysis to efficiently handle instances of the problem when both the total number of candidate actuators and the cardinality constraint scale large. We validate our theoretical results using numerical examples.