I got into arguments using
"Wait for me for a second"
is it actually Wait For me For a second or Wait Me For a second
What is the correct way of delivering the idea behind it?
Taken from Cambridge Dict.:
Wait means
stay in the same place or not do something until something else happens
so it doesn't make much sense to say wait me for a second.
However, it does make sense, and actually conveys your idea if you say
Wait for me for a second.
The page I mentioned also provides a general rule you can use to avoid making such mistakes:
When we use a direct object after wait, we have to use wait for.
So, wait for me(object) for a second is correct.
In casual speech we can omit 'for' before a period of time, so 'wait for me a second' and 'wait for me for a second' are equivalent. Using 'for a second' is slightly more formal but may be felt to be unnecessary repetition of 'for'. Likewise 'I've waited a year for you' and 'I've waited for a year for you.'
I think the right way to say it is "Wait for me for a second", you could also say "Wait for me a second."