For practical purposes, buy a power supply off the shelf. Even if you know how to design one better than the big manufacturers, you can't possibly compete with their cost structure, because they have the volume advantage.
If you have unusual requirments that aren't met by off-the-shelf designs (adjustable voltage, multi-voltage output, constant current limits, etc.), it's probably easiest and cheapest to start with an off-the-shelf power brick or wall wart, and then regulate its low voltage to do what you need. Let someone else deal with the issues around the mains power. That's especially true if you're developing a commercial product that would need agency approval, like UL or CE listing.
But for education and fun, there's no harm doing it yourself, as long as you take the right safety precautions. And it's great to figure out how these things work.
Almost all modern efficient supplies are switched mode, and you need to read up on them to compete in the area of efficiency. As you figured out, they use high freqency, which among other things allows their inductors to be much smaller than the equivalent 60Hz transformer would be.
Without changing things too much, one minor tweak to your current design would help its efficiency a lot and make it run cooler, even if it doesn't bring it up to SMPS standards. Your 5V regulator shouldn't be dropping 20V. Choose your transformer and filter capacitors such that the regulator gets the minimum input voltage it needs to stay in regulation. Depending on what regulator you're using, that's probably around 7V input, but read the data sheet.