Aaand added to my Anki deck 
. I have the feeling that 食べる and 食う work a little differently and maybe that’s why, but I have absolutely no idea about the etymology of either so…
From my monolingual dictionary:
食べる：
固形の食物をかんで飲み込む。
食う：
生命を維持するために、食物を食べる。特に、固形の食物をかんで飲み込む。食べる。食らう。食する。
So while the entry for 食う specifically mentions it’s eating for sustenance, I think they’re pretty much interchangeable - given that 食べる is explicitly mentioned as a meaning for 食う. Regarding the other alternatives mentioned:
食らう：
「食べる」「飲む」をぞんざいにいう語。
食する：
食べる。食う。
So those are just synonyms as well, in terms of meaning, according to my dictionary at least, with 食らう expicitly being very informal.
At the very least, 食う is rougher sounding than 食べる. Probably fine to use 食う if you’re a guy hanging out with friends. Probably not okay if you’re talking to your teacher or boss.
Honestly I think it may just be that it feels weird to nominalize 一段 verbs like that. I mean, 飲み works just fine as a noun.
I was thinking that, but 煮, 似, 起き and 寝 off the top of my head all seem fine. And where would we be without 揚げ!
It might honestly just be that たべ sounds slightly unusual…
I did try googling e.g. 飲み食べ, 立ち食べ, and while the 食い versions are clearly far more common, they don’t seem completely nonexistent. Like someone somewhere spent at least a little bit of time on this graphic:
(though perhaps that’s just an extension of 食べ放題…)
So I think I was just wrong, and you can do this kind of thing widely (at least if you’re willing to coin new phrases), just some iterations happen to be a lot more common than others.
What’s this??? A kind of verb?
Yup.
Ichidan verbs just drop the end to make the stem form like:
食べる 
食べ, 食べます, 食べて
Godan verbs have inflections:
飲む 飲み, 飲みます, 飲んで
The spelling is 一段 (ichidan) and 五段 (godan), respectively. 
Yes
There are two kinds of verbs, basically: ichidan (一段) and godan (五段)
Ichidan verbs always end in る, and when conjugated to the stem form just lose the る.
Godan verbs are some verbs ending in る, and all verbs not ending in る, and when conjugated to the stem form the U sound at the end just moves to the corresponding I sound, basically (so む→み, る→り, う→い, etc.)
There are other differences in conjugation too, but you’ll come across them as you learn more conjugations.
That makes conjugation a bit tricky sometimes. For example 止める is ichidan, so in the polite form it becomes 止めます. 止まる however is godan, so in the polite form it becomes 止まります.
As a general rule (but as always there are exceptions): if it ends in -iru or -eru, it’s probably ichidan.
On the other hand, there are no ichidan verbs (to my knowledge) not ending in -iru or -eru, so any verb ending in -aru, -oru or -uru is a godan verb.
What’s the reasoning behing the terminology? One step and five steps?
段 is also used for “class” or “rank” etc. The ichidan ones have 1 form, the godan ones have 5 forms.
what are, say, the five forms of 飲む? just so we’re on the same page
飲ま (eg 飲まない)
飲み (eg 飲みます)
飲む
飲め (eg 飲める)
飲も (eg 飲もう)
For some added context: the “ranks” here conform to the 5 kana “columns” in the standard kana table (the a, i, u, e and o columns)
So while the stem of an ichidan only has one form, godan verbs have five, one for each column - that really just refers to the last sound shifting across those columns, in the way Leebo just illustrated.
In classical Japanese, there were also nidan and yodan verbs spanning two and four columns respectively, but they’re no longer used in modern Japanese.
Wait what, that’s why they call it godan? TIL
What if I wanted to express this idea of “let’s x” but for 食べる? How do you do that to an itidan verb?
食べよう
Wow thanks riibochan! I thought I was never gonna figure out what itidan and godan were! my (multiple) sensei have taught me some things about verbs but didn’t even mentioned this.
People use the terms u verbs and ru verbs too pretty often I think, and afaik they’re the same thing so maybe your teacher uses that instead?
WAIT, 行くis godan, and it’s a K ending. So is 行きよう right? or is “let’s go” something else? or maybe it NEEDS to be 行きましょう?