What are the essential Japanese counter words?
For example -dai for machines, -mai for papers and stuff like that.
- What else to expect/know?
The article at Wikipedia covers the common ones as well as a decent number of extended ones, and lists the exceptions for days, people, etc. as well as rendaku and number word changes (e.g. 300->san*bya*ku, 4:00-> *yo*ji).
There are entire dictionaries for this (数え方の辞典). Here's a link to a whole bunch. 個 (ko) is the most commonly used one. Japanese people use it often even with words that should have a special counter if it's a less-often used one. It can sometimes be annoying even for them to think of the proper 数詞! Here are most common ones I can think of:
Etc. The list goes on. I happened to live near a harbor so the last one was useful for me (though it was still often substituted with 個!). You can just learn the few most common ones and use 個 for everything else, learning the less common ones as you go.