Given the infinite options, the masters of web animation seem to focus on what looks natural and feels comfortable. Small touches can quickly transform a site from simple, effective but slightly mundane, to modern, exciting and interesting.
Too much movement and it can quickly become confusing and overwhelming.
Two resources that I find myself returning to for inspiration, include:
The 12 Principles of animation is something which I am constantly referring back to. It is a good resource to check whether my animations can be improved.
Google also has some lovely design guidance for making on screen materials feel intuitive and realistic. They set out the rules by which they follow material design within their various applications.
There are various methods for adding animation to the web. Smashing Magazine offers some good guidance of the best approach to take, such as this article. There are also endless examples of developers experiments on CodePen. This animation example using only CSS recently caught my eye.
Some nice examples I have stumbled across on the web include: