Indiekid, are you talking about little metal or plastic cages that enclose the toes of your shoes, with a strap?
If so, in most cases, the clips are pretty well universal, and just bolt onto the front of the pedal cage. The strap threads through slots at the back of the pedal. Provided your new bike's pedals have the holes and slots in the right places, you can switch the clips and straps to the new pedals. The bike in the pic you posted seems to have the right sort of pedals to do this.
If the new bike's pedals won't take the clips (or if it doesn't come with pedals in reality - some don't), you can, as marinyork says, switch the pedals between your new and old bikes, toe clips and all.
Sittingduck is suggesting you get clipless pedals on your new bike. These don't need toe clips and straps: your feet are secured to the pedals by means of springy clippy things on the pedals and cleats on the soles of your shoes. You need special shoes, but the cleat can be recessed so that you can walk normally in them.
Since we often talk about "clipping in" or out of "clipless" pedals, there's often some confusion about which system is which (and many younger cyclists have never seen proper toe clips).
The clipless system needs special shoes, but you can pull the pedals back and up as well as push them forward and down. (Unless you're used to this, though, you won't have the muscles to do it for more than a couple of strokes).
Toe clips can be ridden in almost any shoes, and will hold your feet on the pedals over bumps very well. If you tighten the strap down hard, you can pull up on the pedals, but you can't pull back on them.
Both take some getting used to - in clipless pedals, you need to twist your ankle outwards to disengage; in toe clips you need to remember to pull your foot out backwards (but you already know this).
Hey ho, there's a quick guide to the pros and cons of toe clips vs clipless...
