My first thought was they look like Shimano SPDs as Simon said, but they seem to have toe clips mounted which are a completely different system (from the days before clip-in pedals).
Normally using toes clips you can either ride in trainers, or you need shoes with a metal plate with a groove in it to slot over the pedal cage, however those are MTB-style pedals and shoe plates won't work (even if you could find any these days). I'd just ride in skateboard style in trainers for now - with the straps done up tight you won't have much disadvantage over 'clipless' SPD types except if trying to get started off-road up a steep & slippery hill.