Depends on what exactly you want to do, who your audience is, etc.

Generally speaking, if you want to show something on the web, it's not a good idea to use Flash. You're locking out a lot of mobile devices going that route, as you said.

It sounds like you're asking how you can incorporate animation into your sites, specifically cartoon-like animation? Adobe is just at the start of providing tools for these things, and they're not at the level of Flash. Flash CS6 has an add-on that'll export to HTML, though it is fairly limited in what it supports. They're also working on a tool called Edge Animate, which looks like they're trying to make into the "Flash for HTML" tool. I would look into those as a starting point.

The other thing to consider is if you're just doing cartoons, you could just export a video. You could use any video tool, or actually create the animation in Flash and then export a video file.