Mantra is the sound-form of God-form and is often used synonymously to Devata. It is the means/basis for both Yantra and Tantra.
Yantra is basically a contrivance, charged with the power of a mantra. It could be a geometric shape as in case of devata yantra or just any instrument as in case of an astra. It is the tool. The nature of Yantra depends on its purpose sought to be served.
Tantra is the practitioner’s manual. It combines mantra, yogic methods and philosophy (Tatva-Mantra samanvaya). It elaborates procedures that a sadhaka should follow, at each stage of his sadhana.