The pedal should be under the ball of the foot, never the arch; partly because if the pedal's under the arch your toe can catch the front wheel, but mostly because you have greater efficiency and control when it's under the ball.
Purpose-made cycling shoes have extremely rigid soles to provide support, and provision for fitting cleats that mate to quick-release pedals. You may not feel like going that way just yet - the first outing in cleats can be a bit scary. You can go halfway, though, by fitting a set of strapless toe-clips like
these to your flat pedals. These will keep your foot in the right place and allow you to lift the pedal with your foot as well as just pressing it down. At the same time, it will remain easy to take your foot off the pedal.
With flat pedals and clips you should be OK pedalling in trainers, but if you want to get into serious road- or mountain-biking then proper shoes and cleats will give you greater efficiency and more comfort.
Hope this helps.