well, it depends on a number of things. there are basically two types of core designs. one is the shell type and the other is the core type. in a shell type the windings are wound on a single leg of a core. while in a core type transformer the windings are wound on different legs of the transformer. basically we have to provide maximum flux linkage by winding a number of turns on both the secondary and the primary. then the thickness of wire is selected by considering the current that is to be drawn. there may be tertiary windings on a transformer but that depends on the application.
the high frequency transformers have different core materials that are suitable for high frequencies.
Yes, there are differences between 50-60Hz transformers, audio transformers, and RF transformers. You don't mention how high your high frequency may be.
The core materials are generally different. Power transformers that operate at line frequency can have soft iron cores to magnetically couple the primary and secondary. For higher frequency transformers, soft iron is unsatisfactory because the material has too much "memory" - i.e. it takes too long to reverse the magnetic field when the current in the primary winding reverses. Audio transformers generally use iron modified with silicon, nickel, etc. to reduce the "memory". RF transformers use compressed powder materials such as ferrites, or air - again to get the material to reverse field more quickly.
Winding methods are also different. Power transformers can have the primary and secondary physically separated, but on the same core. Higher frequency transformers such as audio transformers need quicker response to field changes, so they often layer the primary and secondary around the core at the same location on the core. RF transformers require the fastest response, so they are often wound "bifilar" - this means that you take the primary and secondary wire and wind them around the core together at the same time so the wires lay side-by-side. This technique for RF minimizes losses by permiting direct primary to secondary wire-to-wire magnetic coupling.
The below links give some winding theory and suggestions. There are others on the web if you hunt a bit.
50, 60 hz is very low frecuency, so core is not a big problem. We use the same core to build a 50 or 60 hz transformer. The diference is that when you find the number of turns, you need to know the frecuency, so a transformer for 50 hz has more turns that a transformer for 60 Hz.