Does Strionic Resonator cause Medomai, the Ageless to let me take 3 turns upon hitting someone?
Yes, extra turns can stack in this way.

500.7. Some effects can give a player extra turns. They do this by adding the turns directly after the specified turn. If a player is given multiple extra turns, the extra turns are added one at a time. If multiple players are given extra turns, the extra turns are added one at a time, in APNAP order (see rule 101.4). The most recently created turn will be taken first.