bike racing: the penultimate guide: Pedalling 101
It looks simple and it is - and it isn't.
Most of us learn to ride a bike as kids - without toe straps or clipless pedals. (I'm an exception - I didn't regularly ride until I was about 16 years old, with toe clips and straps very soon thereafter.) Years of riding without a mechanically-enforced connection with the drivetrain teaches your legs to push down (and probably to 'grip' the pedal a bit by wrapping your foot slightly around it) but not to pull up. There's no solid connection, so you naturally don't even try to pull up. But as soon as you do get clips and straps or clipless pedals suddenly a whole new world of pedalling efficiency opens up. The big problem is re-learning how to pedal. It's neurological as well as muscular.
Essentially - in my experience only, I'm no sports scientist - you should pedal by pushing down hard from as early in the 'power stroke' as you can, then by pulling across the bottom of the stroke (as if wiping mud off your shoe) before smoothly switching to a 'pulling up' phase which ends just before top dead centre (TDC). The emphasis will be on the downstroke but it's important to realise that the upstroke matters too. And repeat over and over. Smoothly, all the way around.