A website built entirely out of Flash with just an HTML wrapper, is rarely a good idea. Search engines will penalize you heavily, if they can even index your site at all. There are certain types of sites where designing your website entirely in of Flash might be a good idea, but this is usually not the case.
The best way to include Flash animations in your website is with a modular approach. Use small discrete Flash movies embedded in a standard HTML site structure to get the effect you are going for. They will load faster, are easier to edit, and allow search engines to do their job.
Not too long ago I was adding a Flash movie I had created for a client's website but when I went to test it, I got this message displayed as I clicked on it: click to activate and use this control. What the heck? It turns out it is a new security feature in IE. Thankfully it is easily fixed with a short bit of JavaScript. I have included a link to the solution for this problem at the end of this article.
Just because you can animate something doesn't mean you should. This will have a lot to do with your website's audience, but typically less animation is better than more. Try to think like a Japanese interior designer when deciding how much animation is too much. Less is always more.
The speed and smoothness of a Flash animation can make a huge difference in whether it is perceived as distracting or as attractively informative. Fast jerky animations definitely catch your attention, but then tend to immediately put the viewer off, becoming distracting and annoying. If a Flash movie is smooth and can be running in the background while a visitor is looking at another element on the page without being distracted, then you have hit the mark.
Staying at least two release versions behind the bleeding edge is always a smart way to go. The Flash Player is now in its 9th full version, and definitely won't stop there. Macromedia, now owned by Adobe, is constantly adding new features to the Flash Player to enhance the capabilities and fix known bugs. There is about a years lag time before the latest Flash Player is on at least 80% of the systems browsing the web. To make sure that your visitors don't have to download the latest Flash Player to view your Flash movie, make sure you only use features that have been out for at least a year. Admittedly having to download the latest player isn't the worst thing in the world, but why make your visitor go through even a little bit of a hassle when it isn't necessary?
Make it good. Place only quality Flash animations and video clips on your website. Working in Flash can take years to master. Play it smart and hire a pro. Stay focused on what you do best, and let an experienced Flash designer do what they do best. To get quality Flash on your website, higher a designer with a proven track record and examples that you like. Remember to always ask for the native FLA files along with the SWF files when working with a Flash designer.