Learning to count

I'm trying to understand how to count objects, without resorting to translations. So far, I have seen the following suffixes: -hiki, -hon, -satsu, -mai, -nin, -tsu, -dai. Out of all those, I can only infer (when they were used) that -nin is used for counting people. How about the others? Also, what other suffixes are commonly used for counting a certain object? Is there a generic counter suffix, one that can be used with anything?

Thanks in advance.

This should be interesting for you:
Japanese counter word - Wikipedia

Indeed, thank you very much.