Good images are very helpful and sometimes essential in obtaining a useful diagnosis for plant problems. We can sometimes take some educated guesses, but without more information, that's all it would be. The kinds of plants you have is important information for us, as well.

The 'web' might be a sign of a severe case of spider mites, a very common pest of house plants.

We'd especially need to see this 'furry coat' before anyone can help you with solutions.

Being successful at growing beautiful plants takes some study, plain and simple. You need to understand the pros and cons of your particular environment and purchase plants that are suited to that environment. That means, of course, that you should do a bit of research on the plants before you bring them home!

Learning about how to water and fertilize properly is important, as is providing a good potting medium. And then, as you have discovered, being able to recognize the common pests of indoor plants can be very helpful.

When you enter someone's home and admire the beautiful plants, know that it is never by accident. A bit of reading ahead of time would have let you know, probably, that Majesty Palms can be a real challenge in the home. They are frankly not particularly suited as a house plant but continue to be sold as such because they are easy to mass produce and look attractive at the garden centers.

So! How about some pictures of this fuzzy stuff on the palm?

I wonder if the "furry coating" and black spots might be a severe mealy bug infestation with sooty mold from the mealy bug honeydew? Rhizo's right, pics are necessary.