I want to go faster, but I'm torn on which to work on first--smooth spinning in circles or power to push a bigger gear?
Go single speed/fixed gear and you get to work on both (or just pick a gear on your hybrid and don't shift). When I started "really" started getting into cycling I was exclusively a FG/SS rider that mashed on big gears for a daily commute (53x16;87 GI). This meant naturally built in intervals when city riding on the roads. At after a stop light I performed starts like a trackie, pumping the big gear with 80-90% max effort just to get to a comfortable cadence and nice cruising speed (80 RPM was about 20 MPH), but once free from inertia I encountered air resistance and I had to spin quickly to maintain speed to keep up with city traffic. Rinse and repeat at every stoplight.
I did this everyday working day for a year just to go to work and back home. If you're wondering why, it's because I'm not a morning person and I always leave a little late to get to work, thus forcing me to sprint at almost every possibility to avoid being late. Nowadays I'm strictly a road cyclist since I sold my first FG and am currently building two others that aren't in rideable condition yet, but geared bikes are easy mode comparatively now.