Suppose I'm telling a bunch of people to do something. I'm about to tell a person :"You first", can I say to the others :"You second", "you third" ... ? And is there another way to say that ?
The suggestion in the question is fine. You could also point at the people and say, "You first, then you, then you, ..."
Any difference is practical, rather than linguistic. It's probably easier for most people to remember "I go after Jane" and then just wait for Jane to go, than it is to remember "I'm number 17" and accurately count the number of people who've gone. On the other hand, if anyone before me forgets who they go after and I only know that I go after Jane, the whole thing is going to fall apart.
If you want to use other pronouns, then strictly speaking, the answer depends on whether the people you're pointing at are the subjects or objects of the verb. So, if you're describing the line-up at a talent show, "She sings first, he second, he third, then she, ..."; if you're instructing your assassin, "Kill him first, him second, her third, ..." However, most people would probably use "him/her", even when it's not grammatically proper so, in the first case, "She sings first, him second, him third, then her" sounds a bit more natural, to my ear at least. If the verb is left out of all the list items, you can probably use him/her all the time. For example, "In what order are the contestants singing?" "Her first, him second, ..."
If you don't want to get bogged down in the numbers, you can use something like this:
You go first, then you're next, then you, then you, then...
You could repeat "then you go next" for each person. But it's perfectly fine to drop some of the words once you've established the pattern: "then you" or "you're next".
I don't have enough reputation to make a comment, sorry, but in response to the question you asked David Richerby
What if I want to use other pronouns. Should I use objective case or subjective case? Which sounds more natural? "He second"/"him second"
In this case you should use "him second," as this sounds more natural. However if you go with J.R.'s suggestion of repeating the verb, then it would be correct to use "he", as in "he goes second".