The bike shop may have fitted you into a standard racing bike position. That may be OK if you grew up on a bike, but as a newbie, you have to ease into the position. You may be riding with your bars to long and low . Low aerodynamic positions help if you are rididing fast, but if you are cruising longer distances, then a more upright position is more comfortable.

Spinning the pedals is something that you can train at lower speeds. You dont have to go all out to spin fast. Maintain a steady work-rate/heartbeat, drop down a few gears, and increase your cadence. After a while, you will naturally want to cycle a higher cadence all the time.

MTB shoes are fine for road pedals, and make life easier off the bike. Dont feel pressured to change them.

Be careful not to overdo your new excecise. Your muscles build up with weeks, but the rest of your body may take a few months to acclimatise. It probably takes 3 months of regular cycling before you a really ready for hard riding.