Suggesting beginner stuff is easy. Not all of these are what I'd consider the best anime (some of them are awesome), but they're all good gateway anime for their respective genres in my opinion. For these I'll try to pick things that don't rely heavily on a knowledge of the medium to enjoy.
Beginner
Mech/Space Mech - Gurren Lagann. Even without a knowledge of mecha, Gurren Lagann is incredibly enjoyable on a base level.
Sci Fi - Steins;Gate. This one I was more unsure on because it does throw a lot of Japanese pop culture at you, but I think the story itself is more than compelling enough that they won't be alienated by it.
Slice of Life - Toradora. This one because I see it as a kind of representative for all of the slice of life high school stories that are out there. May as well have them see a good one before they realise that genre is oversaturated to the point of breaking.
Fantasy - Fate/Zero. This might be controvertial because it's a prequel, but it was my first exposure to the franchise and I think it is the superior series.
Comedy - Nichijou - This one because it has a kind of manic energy that should make it accessible to all. That and I also want people outside the fanbase to realise it isn't just memes.
Drama - Eve no Jikan - Tricky one. I don't really think there is a whole lot of anime that I'd describe as drama (far more that fit the definition of melodrama), but I recall this one got fairly heavy at certain points.
Action - Fullmetal Alchemist Brotherhood - I can't not put this as the action anime for beginners. It is so famously a gateway anime.
Sports/Hobbies - I'm not too big on this genre, but I liked Chihayafuru, and I think that one is fairly easy to get into even if you don't watch much sports anime.
Game/MMO - Well the obvious one for beginners here is SAO, though I don't know about this being its own genre.
Romance - Clannad, because it's hard to find many high school anime that has a romance beyond a first kiss, and I think this is probably the best popular example I can think of.
70's - Galaxy Express 999 - I'll fully admit that I haven't watched a lot of stuff from the 70s (I know, shameful), but out of the stuff I have seen, this one seems the most accessible.
80's - Dragonball Z, because even though I am not the hugest fan of it myself, it is still one of the most popular gateway anime even now.
90's - Cowboy Bebop, because this is a good anime that even people who don't like anime enjoy. Therefore I think it fits in nicely as an anime for beginners.
Post-Millennium - Attack on Titan, since a lot of mine are post-millennium anyway, and I would be remiss not to mention this since it has caused more people to start watching anime in the last few years than I could possibly fathom. This is another one that made the transition to the mainstream, and is enjoyed even by people who normally dislike anime. I don't personally think it's that amazing, but I can't deny its impact.
Now, I don't know how good my veteran suggestions will be, but I think I've watched enough more obscure/underrated stuff to have a crack at that too.
Veteran
Mech/Space Mech - Martian Successor Nadesico, because it is oddly underwatched, and is a wonderful ode to mecha in general. I don't think someone who hasn't seen much mecha would quite get it.
Sci Fi - Casshern Sins, because it fulfills the requirement of being both underwatched and really quite good.
Slice of Life - Genshiken (S1, S2, and the OVAs, less so Nidaime). This one is my favourite ever slice of life show, and I think a great deal of it would be lost on a beginner.
Fantasy - I can't think of anything obscure enough here. Sorry.
Comedy - The Comic Artist and his Assistants, because for my money, it is one of the stupidest things I've ever watched and is way underrated/underwatched. There is nothing remotely intelligent about it, it's just every awful anime trope you can think of except in overdrive. Somehow it works, I don't know why.
Drama - Legend of the Galactic Heroes, because it's 90% political intrigue to 10% space battle. This one might not be that obscure, but it's recently become available via legal means, so I thought I'd mention it.
Action - Rin: Daughters of Mnemosyne, because it's got a lot of the bizarre stuff that makes me love anime. It can be a little bit shocking on occassion though, so fair warning.
Sports/Hobbies - Chihayafuru was all I really had, but if we stretch the definition a bit, I'll say Akagi was incredibly entertaining to watch even though I still have no idea how to play Mahjong.
Game/MMO - Log Horizon. Oh I don't know. I just think if I have to pick something specifically in this subgenre, Log Horizon was the one that did it best in my opinion.
Romance - Maoyuu. This one is often just seen as derivative of Spice and Wolf, but I found it to be a very enjoyable anime that is underwatched. I really wish there were more of it.
70's - Belladonna of Sadness, mostly because it's really weird, quite a visual spectacle, and really obscure. Although it has become a lot less obscure since AL released their Blu-ray of it.
80's - Angel's Egg, just because like many pre 90s anime it is ruddy hard to get a hold of legally. Seriously, somebody needs to do a re-release of this ASAP. I'd pay more money than is wise to have an artsy collector's edition of this movie.
90's - Golden Boy, because it's Golden Boy. Anyone who has seen Golden Boy will understand what I mean. I had to get it on the list somehow.
Post-Millennium - Samurai Flamenco. I thought since this category is so broad I'd just use it to mention that I love Samurai Flamenco and you should too.
Basically fighting an interstellar war with the power of rock music.