WHEN I code a site, I prefer to use as much plain HTML as possible. This way, I feel more certain that there are no barriers to anything I write because plain HTML is the "lowest common" technical requirement to use the Web.
However, plain HTML can be quite dull. If you look at the home page of user:number 1, very little happens there to keep people watching. That's fine - the site is primarily an information website and was designed for visitors to find information quickly.
There are simple rules-of-thumb to be guided by here. The most important thing is not to annoy your potential customers. This means that any transitions should be kept to a minimum. Keep down the total size of the page - many people still use dialup connections and a fast-loading page is always preferred, all else being equal. Finally, ensure that the transitions add to the experience rather than hindering them. So for example, animations can be used to draw the eye towards relevant information; but they should also not be distracting (this fine line is possible - I tested this kind of thing myself in the 1990s).
Testing is the best way to understand where these barriers are for most people. It's possible, but also risky for a designer to rely entirely upon their own judgement - their page load times are probably good and they're being paid to do this. Customers aren't - they are paying for the privilege.
There's no pleasing some people ;-)