I have several DIY power supplies. They use either the 78xx or 79xx chips or the 723 which allows adjustable outputs. I have used them for decades now and they have been perfectly adequate for most of the work I have done. The one I have used the most is a four output that uses only one transformer. The two positive outputs use 723s and are adjustable so I can have from around 2 V to a bit over +15 V. The two negative outputs are -5/6 V and -12 V and use 7905 and 7912 chips respectively. The -5/6 V is switch selectable by adding a pair of resistors in the ground connection of the 7905. I did this for designs where I wanted a +/- 6 Volt supply. These ICs provide short circuit protection so accidents have not caused any damage over the years. The balanced outputs have been handy for op amp circuits that I have designed a lot of.
Building linear, regulated supplies is not difficult. I just followed the manufacturer's data sheet recommendations.
But, today you can purchase a lab style supply for a very reasonable price. It will have both Voltage and current metering and all the protections. I could not build a supply, even without the meters, for the price of an already built one. I do have a nice one that cost about $50 USD and will probably buy more as time goes by.
You mention a safe approach to mains wiring. Generally speaking that is not that difficult. My present approach is along these lines:
1. You can purchase power entry modules that will have a fuse holder built in. The wall cord will plug into them. This is a very safe way of getting the power into your chassis. They come in a variety of wiring styles: I usually get the ones with solder lugs.
https://www.mouser.com/c/connectors/power-connectors/ac-power-entry-modules/?fused=Yes&termination%20style=Solder%20Lug~~Solder%20Terminal&rp=connectors%2Fpower-connectors%2Fac-power-entry-modules%7C~Termination%20Style2. The power cords are available with the mating connector already installed. They are commonly called D-pin power cords.
3. I use a completely enclosed, metal (aluminum) chassis/box to enclose the power supply. Aluminum boxes or chassis are commonly available and easy to work with (as opposed to steel). DO use a metal box because it forms the primary protection against accidental shocks. The mains wiring is entirely inside the box except for the power cord and if you use a pre-assembled cord it will meet all necessary safety standards.
4. Good wiring practice from the power entry module to the power switch and transformer should be followed. This is normally point to point soldered wires. Although not normally required, heat shrink tubing can be used to insulate the solder connections.
5. The safety ground is connected from the power entry module to the chassis with a short ground wire. I normally use a crimped ring style terminal to one of the mounting screws for the power entry module. A star style lock washer should be used between the screw/nut and the terminal ring. That provides the return path to trip the circuit breaker in the building's breaker panel in case of a short to the chassis/box. This is a very important step.
6. For most of the power supplies I have built a 1 Amp, fast blow fuse has been more than enough. A 12 V, 5 Amp supply would be rated at 60 Watts and at 115 V that would be only about 0.52 Amps so that 1 Amp fuse value is OK. The fuse is installed in the power entry module. However, if your supply is for more than 100 Watts of so, you should actually calculate the fuse size needed.
That is pretty much it. I like to mount the transformer near the power entry module. The power switch is usually at the front. One of the transformer leads can usually be soldered directly to the power entry module. The other one often must be extended to reach the power switch. Heat shrink tubing can be used over an inline splice. I like to use a double layer of the heat shrink tubing (two sizes, one over the other).