Getting older and slower!
Join Date: Feb 2008
Location: Bowling Green, Kentucky
Posts: 419
Bikes: Trek Domane 6 series Project One, Bike Friday Pocket Rocket, Trek XO1, Specialized Turbo Vado 6.0
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You are correct you want to spin. Not only will you get more cardio, you will eventually ride further without fatigue, ride faster, and climb better. And you will also develop a smooth pedal stroke.
I assume you have a bike computer that measures your cadence. If not, I recommend you buy one. Then ride focusing on your cadence, not your speed. (I do this the first 500 miles or so I ride every year.)
Focus on increasing your cadence gradually, like 10 rpms at a time. And keep track of your progress. The record will hold you accountable and give you honest feedback towards your goal.
Good luck!