I'm a rabid, er, uh, avid cyclist and can shed some light on this for you.
Serious cyclists who participate in spinning classes often bring their own shoes and pedals with them. The foot retention system you choose is a matter of personal perference.
Some systems, such as traditional slotted cleats and toe straps, lock your foot into one place on the pedals and if that place isn't right for you, you can get sore knees pretty quickly and damage if you persist.
Toe clips and straps, combined with plain old flat pedals and sneakers of some sort will allow a little bit of movement but placement is still important.
Many modern pedal systems, knowns as 'clipless' systems because they don't use toe clips, attach the shoe to the pedal with some built in float, almost like a single point was attaching the bottom of your foot to the pedal but it wasn't tightened. These range anywhere from a few degrees of float to 20 or more with Speedplay and a few other systems. Some even allow the amount of float to be adjusted.
I suggest you try the clips and straps at your spinning class and see how they work for you. Most cyclists find that keeping their foot relatively far forward on the pedal, with the ball of the foot directly above or a little in front of the pedal spindle (its 'axle'

provides the most comfort, while bringing the foot a little further back can be more efficient, albeit sometimes at the expense of comfort. (Be aware that your fore-aft foot position on the pedal is important and that moving your foot forward on the pedal effectively shortens your reach to the pedals and thus affects what your saddle height should be.)
One final tip. Experienced cyclists usually think about pulling across the bottom of the pedal stroke, as if scraping mud from the bottom of the shoe, rather than pulling up on the back stroke. For whatever reason, this seems to work better.
Hope this is of some help.