Using cycling shoes with toe-cage pedals (like the ones you have currently) works fine for most types of cycling shoes, but you may want to check compatibility before you buy. Cycling shoes have rigid soles that will help you transfer more energy into your pedals, helping you move further with the same effort.

Many people nowadays will prefer "clipless pedals", which are actually pedals you click into (many discussions on this forum to explain the name paradox, check out the archives to know why). They are also called "automatic pedals". Cycling shoes have tiny holes under them for you to attach cleats (compatibility issues arise here) that attach to the pedals. Very popular pedals are Shimano's SPD pedals for touring or mountain biking, which have recessed cleats so you don't walk funny (unlike SPD-road pedals, Zero, Look, etc. whose cleats are protruding outside of the sole). I also find them easier to clip in and out of. Great for recreational riding, commuting, etc.

But, again, many cycling shoes will totally work in toe-cage pedals, and will indeed give you a bit more strength. I had a pair of touring shoes (they looked like hiking shoes, only with rigid soles) and sometimes I'd use them on my commuter bike, which has regular platform pedals, and I would notice a big increase in power-transfer. So if you don't want to invest in shoes AND new pedals, you can always just find a pair of shoes that fit your toe-cages and get going! In the short term you may want to invest in SPD pedals, which are 30-50$ (for an entry-level, VERY decent pair), so make sure you buy shoes that are SPD-compatible.

enjoy the great world of bike accessories!!