Hmm… this is gonna be a hard one.
For starter anime, Attack on TItan and Fullmetal Alchemist: Brotherhood are good choices. While I haven’t watched them, I’ve heard that Studio Ghibli films (ex. Spirited Away, My Neighbor Totoro) are good gateways to anime.
If you wanna see an anime with over-the-top action, check out Gurren Lagann or Hellsing Ultimate (though Hellsing Ultimate is pretty gory).
Speaking of familiarity with western shows and a preference for comedy…I wouldn’t consider this a good recommendation, especially to people unfamiliar with anime, but Panty & Stocking comes to mind. The art style is like that of western cartoons, and it’s also a comedy (though it’s mostly vulgar humor). However, it’s also fanservicey, so you probably won’t like it. Still, it’s my second favorite anime, (the first being Gurren Lagann, and the third being Hellsing Ultimate) and I’d recommend watching it once you’ve gotten familiar with other anime. Also, the show has a great dub; I'd recommend watching it over the sub.
Space Dandy’s a pretty funny comedy series.
As for sites, I like Kissanime. It’s got both dubs and subs (though some dubs might not be available), and there aren’t any video ads. Also, the ads on there aren’t obtrusive, and can be hidden by a button below them, so you can disable adblock on there (the site has an anti-adblock script, but considering what I said, you don’t really need to use adblock on there).
Animehaven's also a good site with high-quality streams.