You should definitely watch
Tokyo Godfathers around Christmas (it's a movie rather than a series)! It's by the talented Satoshi Kon (who also helped in stuff like
Paprika,
Millennium Girl, and
Paranoia Agent, the series) who passed away some amount of years ago, iirc. It's both Christmas themed, warm, and a bit unusual with their selection of characters. It's a really sweet movie, I think.
As far as series go, I generally suggest
Tatami Galaxy (Slice of Life/Drama) because it has some metaphysical aspects to it (and even some magical realism; an entire subplot/concept is lifted directly from 100 Years Of Solitude) without being too polarising due to the coming-of-life schoolage aspect to it (but I feel like this is a theme in most anime lately...). There's a bit of comedy to it, too. It's in somewhat the same vein as
Welcome to the NHK! but it has a stronger female character in it, I think, and it's a bit more versatile despite the Groundhog Day effect of repeating the same span of time. If nothing else, the person that recommended that should check this out!
Baccano! seems like a good start because it has a little bit of everything; an intriguing plot, somewhat endearing characters, accessible comedy, and it follows different characters around, so if some of the protagonists bother you, hey, you'll eventually get to see some others instead.
Madoka is an alright series, especially since it takes a well established, pre-existing theme and kind of inverts it, but I feel like it'd be more appreciated by someone exposed to magical girl fare first (
Cardcaptor Sakura,
Sailor Moon, etc.).
A nice handful of series here are classics, and the only reason I'd recommend against them is because it's generally pretty fun to go out and explore new offerings.
Cowboy Bebop (admittedly a personal favorite),
FLCL,
GTO,
Lain,
Akira, etc. are all pretty solid series, but I feel like a lot of people have probably seen these before because a) they've been around for a pretty long time and b) they've also been broadcast in the US, sometimes even with dubbing. Definitely worth watching any of them, though. I feel like Lain is a bit underrepresented sometimes, for example. Oh,
Samurai Champloo and
Trigun also fall here I guess, I'm just not really familiar with either of them to any meaningful degree so I feel it'd be fallacious to assert either are good.
I also feel like some of the series suggested are pretty polarizing and perhaps inappropriate for application in this particular manner; stuff like Lucky Star, Yuru Yuri, K-ON!!, and even CLANNAD are kind of poor choices to start with since a lot of the characters are fairly one dimensional and ultimately not terribly compelling as far as storylines go; it's all farcial emotional stuff (and in Lucky Star's instance, a lot of anime references that not a lot of people will appreciate unless they're well versed or using a guide or something) and fairly bland stories. S-sorry. I also feel like Princess Jellyfish is in a similar vein due to the Amars and their general misandry, but only from threads I read about it. I kind of enjoy the series, but the ending was a bit weak, and from the suggestions, there are better things to start with. Nichijou is also pretty good and I like it a lot; it's varied, it's a bit dynamic, kind of a composite of vingettes, but some of the humor is a bit weird/drawn out, I think? I like it as a series, but feel like there are better things to start out with, again.
Series that I altogether dislike: Higurashi, Mirai Nikki, Shingeki no Kyojin on the merit that they're mostly slaughterfests and pseudointellectual, sort of like how I feel about Elfen Lied. Higurashi in particular seems weird outside of a VN format, since it has to adopt this groundhog day quality to it with the protagonist hanging out with different characters which seems kind of obnoxious and unnatural, a necessity of adaptation rather than a thematic element of it. Ergo Proxy is probably good, but I found it dull to the point of being offputting.