Working with building, upgrading, or repairing computers can lead to great technological victory or humiliating defeat depending on a lot of factors. Using your list, it is possible to see some common issues that one would need to be aware of before attempting any of those three procedures. Proper static protection is important for all of these, so lets just get that out upfront.
1) Power -- the biggest problem with the power supply is making sure that it is powerful enough to run the components attached to it. You would, if building a computer or adding components, want to add up their consumption to be certain the power supply could handle it. Secondly, if a power supply is "dead" and not responding it is a good idea to check for an internal fuse that might need replacing. Technically there is a voltage switch on the back of many power supplies that might have gotten flipped as well, though that doesn't happen too often. Lastly, if putting in a new power supply you would need to be certain that it fit the case and had the connectors you needed to properly drive the motherboard (especially video card power.)
2) Video -- First, make sure you have the right card for the slot: AGP and PCIe are not compatible. Some motherboards seem to have problems with backward compatibility, too, between 4x and 8x AGP so that is good to check into. You need to also consider the size to make sure it will physically fit in the case. For full functionality, many video cards require an direct power hookup to the power supply, so that needs to be checked. Make certain the card is seated well in the slot and that the correct drivers are installed.
3) RAM -- The RAM has to be right. Though there is a good deal of backward compatibility, you need to be sure you have the right stuff or it won't fit (or maybe it will fit and won't work, if it is too slow.) Some RAM needs to be installed in pairs, period, or in pairs for better performance. It is important to make sure that RAM is well seated in the slot or it will cause you trouble. Troubleshooting RAM usually involves swapping it out, changing it to a different slot, or pulling it out and running with less RAM to see if that stick is defective. There are software solutions, too.
4) CPU -- Again, it has to be the correct one to fit in your motherboard. Must be installed carefully so as not to bend the pins. Cooling is very important, so the heat-sink/fan must be the right size and properly installed. There may be some BIOS settings involved with setting up the CPU you way you want it, but in general it will be self-configuring. If it doesn't work it may be damaged or improperly seated....if the computer shuts off or freezes heat may be an issue.
5)Hard Drive -- Fairly straight forward. Make sure you have the right connectors ATA/SATA. Make sure the jumpers are set correctly. If it doesn't work, check the power, the connection to the motherboard, and the jumper settings. Hard drives are pretty tough when not turned on but easy to damage when powered up and active, so don't jostle it. I don't know much about SSDs, so you are on your own there.
6) Optical Drive -- Same as the hard drive, really. Jumpers and cabling. If you want to install a Blu-Ray drive you have to keep a thought out for HDCP compliance between it, the video card, and the display.