Movies aren't my strong side, so I'm just listing 20 ones that I feel are the most essential.
Here's how the series chart works:
This is the genre hub. If you like fantasy - start with FMA:B, if you're a fan of Firefly/Guardians of the Galaxy - go for Cowboy Bebop, if you want something with a villain protagonist - watch Death Note, and so on. But overall these 12 series are a good brief look at different types of anime - all beginner-friendly, so some of the more serious and hard to digest stuff is hidden deeper. For now, try all the starters out.
If you liked the starter, check the color above it. Below, there are 12 sets of 12 anime belonging to these particular genres. For example, if you've loved Toradora and want to check out other romance anime, scroll down until you find an image with pink background. That will be the romance section.
Inside a section, the anime will be listed in three rows. First row will be always anime that are the most similar to what brought you there. So for One Punch Man it will be superhero shows on top, then below two bit different types of shows with superpowers. Also, within a single row, the most noob-friendly anime (in my assessment) are placed on the left side, and on the right side are the more challenging ones that might not be as easy to get into.
Before you ask why there's no Big 3 in battle shounens - these series are best consumed in manga format, the anime adaptations suffer from pacing issues (and filler). In the time needed to binge hundreds of episodes of one of them, you could finish dozens of other series, some of which you may find more interesting. Berserk and Claymore are another read-the-bloody-manga cases, but they need and deserve the recommendation and their anime are watchable.
NOTE: "Seinen" doesn't mean "blood" or "slasher horror". There are many seinen series (for adult male audience) from outside of the action genre - comedies, romances, tons of slice of life series.
As far as magical girl section is concerned... LWA is a Westerner-friendly decent starter, but if Cardcaptor Sakura is too old and cutesy for you, you should consider starting with Madoka (you will find it in another section). That's bound to work.
Basically the less you know coming in the better. Thrillers and horrors are rare to come by, so if you're a huge fan of the genre you're going to have a hard time sating your hunger for more - but there is a shortlist of worthwhile stuff.
In this chart, I've made the drama section about "feels" shows (top row), cathartic introspective shows (middle row) and "patrician taste" shows (bottom row). That left out some worthwhile low-key drama shows like Hibike! Euphonium, Rainbow, and plenty of others, so please do not limit yourself to this chart if you wish to dig for true gems.
Please don't take the "for people with taste" phrase too seriously, I haven't finished a single show from that row either. Have some distance.
Nisekoi is pretty rage-inducing, but the reaction faces make it funny enough to put it as the noob-friendly option. Please drop it if you demand any romantic progression and move on to the other suggestions.
This "genre" block is a mix of action, comedy, and ecchi - basically if you want to watch anime for the PLOT, this is your jam. Have fun. There's tons of other highschool battle harems available and some aren't even garbage, so you're going to have tons of it.
Bottom row here is a bit lower on ecchi, but has some of the most batshit insane scenarios ever written. If you ever thought "anime is crazy"... You have no idea.
These anime didn't belong to any genre blocks, at least not entirely. If you're completely new to anime, you should ABSOLUTELY AVOID THEM (aside from FLCL, I guess) as much of the value will be lost on you if you don't know the genre basics they're about to shatter, or are too overwhelmed to understand the symbolism and the underlying messages (you'll need a "WTF did I just watch" guide for half of these).
You might suspect that there's a reason they're on the chart anyway - well, suffice to say they are some of the most thought-provoking, discussion-sparking, shitpost-generating anime ever made. This is the end of the rabbit hole. Welcome to the weeb squad. There is no exit.
Shoutout to all folks from /r/anime's Discord who helped me polish up this chart. You're awesome.