There are plenty of tutorials for this around, but you should even find pretty good explanations in each program's documentation.
Basically, if you have Flash yourself, you will probably not want to export as .swf. You'll have much more flexibility when creating that .swf in Flash instead. It's also reusable, if you for instance create a favourite design. The .swf is the player, the 'skin'.
The FLV video file (which you will always need in addition to an .swf - they can come from different sources however) can be made in either of your programs. The 'science' here is choosing a good size and compression rate which will give optimum quality with minimum download time / lag for the user. Start with a preset, and learn as you go.
Upload the .swf and the .flv to your webserver, in the same folder probably. Insert only the .swf on your webpage, and you are ready.
Not everything that can be counted counts, and not everything that counts can be counted.
What are your export options in final cut? I'm more familiar with Premiere, but you should be able to export into at least MPEG and from there use a video converter to get your desired .mov, .flv or .swf format. If your on a mac I would very highly suggest Visual Hub and as a not so close 2nd, Roxio Toast. I just think although Roxio is more elegant, it's not as simple or fast as Visual Hub. For PC there are ALOT of applications that can accomplish this. I have been satisfied using Wondershare Video Converter, but I'm sure there are plenty more for the PC out there that I don't know of