For a computer, it is very true that turning it on is the hardest on the components and drive. I worked at a very large company and we left every computer on 24/7, except for weekends. In just the home office, this would be around 500 computers and 200 dumb terminals. We never turned the servers off except to shut them down once a month, and restart them. But, I wonder about the more sensitive devices these days that are completely digital. The analog world had no issues at all with any of this, but digital chips are very sensitive to surges in current, spikes, fluctuating power, etc.
Some things would be perfectly fine to leave on all of the time, except you are paying for the electricity. And digital things don't take much current to leave on. Just be sure they aren't on the same breaker as your window AC unit or a refrigerator. When the compressor comes on or goes off, in one of those, a surge is sent down the line to everything and over time, it will damage your digital chips.
A computer on the same line would be the same, since it is primarily digital, but never turning it off is also not good as you might not get the full advantage of your RAM unless you do. Some programs don't give RAM back, when you don't use them. So, the only way to get all of the potential of your RAM back is to do a warm boot or cold boot. A warm boot is more gentle than a cold boot. Also, just turning it off isn't a good thing, either. If you have programs open, they can be damaged by files being corrupted, because they were open at the time power was removed. And even if you just have Windows open, you can create digital holes in your hard drive called bad sectors. Removing power without shutting down may create lost chains in that line of sectors. Usually, the computer can repair these lost chains with no problem, but sometimes you end up with bits of files with no home. You can't put those file bits back where they came from and you've lost data.
If you have power creators like amps or something that sucks a lot of current on the same line as something digital, they will also be OK to leave on, as long as you want to pay the electricity bill. But, some of those can get pretty hot. Heat is not good for digital chips and if you have tubes, they have a life span that you are using up by leaving them turned on. Granted, there is no set time on tubes, but just like light bulbs, they can burn out at any time.
If you want to turn your equipment off when not in use, if your equipment has a good surge suppressor in-between each, I don't think it matters that much about a sequence of turning things on or off. But, if you don't, I'd suggest turning things on and off in a set sequence. Turn power creators like amps on first, and off last. Turn any really sensitive digital things on after amps are fully up and turn them off before you power down your amps.
If you have any devices that load drivers from the computer, you'll have to have them turned on before you power up the computer, but that shouldn't be a problem. Just don't assume that those power creators in your system aren't pumping a surge into your sensitive digital stuff. They can act just like a compressor in a window AC unit or refrigerator. Surges in current, uneven current, and spikes...those are some of the worst things on your digital chips. This process will give you less power surge on the line when turning the high power devices on and off that might send a surge of current through the line to any digital device that's turned on.
On a side note...
Many of you know that a hard drive will place parts of a file where ever it finds space. Music files can be quite large and unless you have sufficient drive space, the computer can end up putting parts of a file all over the place, where ever it finds room. When you call that file back up again, the computer puts all of those bits back together again before it's a whole file. Large drives with a lot of space will do this a lot less than a drive that has limited space for new files. Defragging your drive is a good practice to get into, even if you have sufficient space for flies. It just keeps your drive happy and doing less work. It will also slightly speed up the loading of large files. And if there are any sectors that have an "iffy" surface area, defragging will find them and either repair them or the computer will mark them as unusable. That's a good thing, but I don't know if I would continue to trust a drive that has any bad sectors that weren't repairable.