Doc : In vino veritas.
Ringo : Age quod agis.
Doc : Credat Judaeus Apella, non ego.
Ringo : Juventus stultorum magister.
Doc : In pace requiescat!
Doc : In wine there is truth.
Doc Holiday is excusing his own behavior here, and further insulting Johnny Ringo by saying that he is drunk, and saying truthful things he would otherwise not reveal. He had previously said he hated Johnny Ringo for being similar to himself.
Ringo : Do what you do / Watch what you do.
This is one of the most interesting lines because it means more than just watch what you do. The line can be interpreted as be careful, or people do what they do (saying that Doc Holiday is drunk because he is a drunkard), and it can also mean something along the lines of do what you do best, which would be gunfight since Ringo had apparently heard of Holiday's skill. It is a challenge and an insult combined into one.
Doc : Tell it to someone else, not I.
This line is dismissive. Doc Holiday is conveying the fact that he doesn't care what Johnny Ringo is saying and that he doesn't care what his advice is.
A common Latin saying meaning "Let the Jew Apella believe it; not I". The phrase means, roughly, tell it to someone else, not me.
The reference is taken from the work Satires (book one, satire five) by ancient poet Horace. It is derived from a scene where people try to convince travellers of miracles happening at their shrines. The phrase is uttered to convey the disbelief and that they should tell their stories to someone else.
Ringo : Youth is the teacher of fools.
When Ringo taps his pistol he says this, which conveys the idea that Doc Holiday is inexperienced (youthfull) and ignorant of the danger he is getting himself into.
"Juventus stultorum magister" is a common Latin aphorism, or phrase that has many implied and implicit meanings. The sentance translates to "youth is the teacher of fools".
Juventus isn't really a latin word, they didn't use the letter 'J' so it is really Iuventus, which is in the nominative case (subject of the sentence), and it means youth or adolescence.
Stultorum is a latin derivation of the word stultus, stulti which means fool. The "orum" ending places it in the genative case, used (most frequently) to show possesion. Hence of fools.
Magister literally means teacher or schoolmaster. It is in the Nominative case as well.
There is an implied transitive verb such as "is" which would require the nominative case on both sides of the verb. Hence we have: "Youth is the teacher of fools".
Doc : Rest in peace!
To end the conversation Doc Holiday throws the previous warning back into Ringo's face. Doc tells him to rest in peace, or to die, because Ringo is unaware of the danger that Doc presents.