First rule is one of the most trickiest to obey and it is this: don't touch the clock too much with your hands. The gold plating on the pendulum, weight shells, dial and movement get eaten up by the acid from your hands. They will end up with black marks wherever hands touch it. This is not to scare you into not touching it at all, but just to limit the areas you touch, like try to hold the movement by its edges rather than a noticeable spot like the back plate or the shiny area of component such as the dial. Cloth gloves may also be used.
When a pendulum clock does not keep time in a consistent manner, such as losing or gaining 5 minutes a day, it is adjusted by the nut at the bottom of the pendulum bob. Turn the nut to the right to raise the bob and increase the speed of the clock, or the opposite way to lower the bob and slow time. Turn a couple of revolutions with the nut only, otherwise it will go to far and you will have a hard time getting back to where it was if the clocks time gets way out of wack.
If it comes down to looking at the clock movement, take off the weights and the pendulum before moving the clock case. This is so the weights don't swing to far and trash the glass in the clock case or the pendulums suspension spring doesn't break off. So remove the weights and pendulum even if your only going to turn the case to get to the back of it.
Some weight driven clocks are chain driven and some are cable driven, the text is covering more chain then cable but the same principles apply to both unless specified.
Note on replacing rather than repairing
If your clock was made from around 1965 to the present day, there is hope to get a new movement to replace your old one. Replacing is better than repairing because the new movements are free from bushing wear and usually the maker of the movements improve them over time. To pay someone to overhaul a movement that is still made does not make sense because the cost would be about the same to get a brand new unit. We often charge even more to overhaul a movement then we sell new ones for. This is because of the time and effort involved with the overhaul.