I forgot to recommend AnoHana and Kowarekake no Orgel because they are not as "harem" as Amagami and Clannad, but I consider them very good anime also and second their recommendations.
Fate/Zero is an amazing series, although I'm still on the edge about whether or not you should "know" the
Fate/stay night anime/plot before. Although F/Z is technically a prequel, it airs after F/sn. I find that by knowing what will happen from watching F/sn first, I'm able to focus more on the intricacies of F/Z and appreciate it more. So yes if you plan on watching F/Z I'd recommend F/sn first.
Since you want to know about Clannad/Kanon/Air, I have a small comparison for you with light spoilers, so click if you don't mind.
Spoiler for Light synopsis spoilers for Clannad/Kanon/Air:
Whether or not to recommend Kanon/Air depends on what you enjoy from Clannad. For example, I like Clannad because it brings many elements together that do not work in a typical romance, so when I try to think of what other anime accomplishes this, I can only think of ef and Bakemonogatari.
Clannad vs. Kanon
If you really liked the witty protagonist Tomoya and the jester-like Youhei, you're not going to find that in Kanon. Kanon in contrast a more serious anime in a sense that it has a consistent gloomy/serious mood (which is good) without the usual comedic interventions that Youhei gives to Clannad. Another thing is that Kanon arcs are more definite. For example, after Shiori's arc, you won't see her again (until the last episode). In contrast, Clannad characters stay in Nagisa's club, so they still play a role after their arc. So overall, if you like Clannad most for the drama in After Story, then you'll like Kanon, probably.
Clannad vs. Air
Air is something else. It's more story-driven and to the point. It follows the general trend of lighthearted -> drama progression as Clannad, except it does so in 12 episodes instead of 52 or so. So in short, it doesn't fuck around. You won't get a season to ease you into drama, it just happens. The storyline is more convoluted but that's not a bad thing. You won't have many opportunities to laugh in Air but it's not depressing either...in fact arguably the opposite. It's just in my opinion not that similar to Clannad at all.
Conclusion
What Clannad, Kanon, and Air have in common is the addition of fantasy overtones that do not drown out the realistic slice-of-life nature of those anime, although the realism could be argued in Air. I like all three anime (Clannad: 8/10, Kanon 2006: 7/10, Air: 9/10). They are also similar in a sense that you feel like you got something out of watching them, and it gives a new meaning to how you view anime. But structurally, I feel the most similarity between Clannad and ef, probably because it comes down to the characters and how they interact with each other. /endtldr