I can’t say for sure this would win any awards, but here’s where I might start. Look at a website that specifically tries to appeal to everyone, and understand why they make the design decisions that they do. Unfortunately, for a lot of typical sites, the design is driven by the desire to sell something (advertising perhaps). That’s why I’d say Wikipedia is the best for this purpose. They’ve even got guides on how and why they format articles in particular ways—and a lot of their policies are guided by compatibility and accessibility.
So, for example: the site must fail gracefully. Go ahead and use fancy features, as long as there’s an alternative. For the most part, don’t even prompt the user—when something goes wrong, just switch to the alternative and give them the content.
Support old browsers and mobile browsers. A tasteful notice about degraded content is fine, but certainly don’t interrupt the user with a modal dialogue box.
Structure it in a way that allows alternative access like screen readers. Wikipedia has an interesting take on formatting for blind users—they prefer accurate metadata in their images, so that screen readers can say something useful about pictures, rather than reading HTML tags aloud.
Make it fast, and keep it available. Nobody wants to wait for content. This means picking a good web host who will devote the appropriate resources to keeping the site up.
And don’t require Flash. Especially not for a splash page. Just forget about that. (With mobile platforms being quite popular, and as ever, a significant minority unable to use it for technical reasons, and plenty of alternatives available, why bother with something like that?) It’s alright to use Flash for video, but failing gracefully is the key. Maybe offer a link to the audio as well, so that the content is still accessible without video.
Content is king. If the site is nice, but empty, your chances will be reduced. Make the site useful, and something people naturally want to visit, and the website judges will naturally appreciate this.
Finally, plan for contingencies. Sooner or later, you’ll move on. Make sure it’s obvious to the next webmaster how the site operates. Keep records and document your conventions, databases, passwords, etc… Also, keep occasional backups. It’s boring, but necessary. (You’ll look like an idiot if you have to tell the team, “sorry, we lost everything…I’m going to go try the Wayback Machine”.) Your webhosting provider may be able to arrange something for you.
(By the way, I can claim a familial connection to a whole slew of website awards, including the big one, because my brother designed and webmastered 188’s website for a number of years. Some of this draws upon what worked for him.)