The Latin, Greek and Knox translation can be found on the New Advent page. You can listen to the Latin here (from 1.38) and the Greek here.
Here is the Latin:
6 Modicum, et jam non videbitis me; et iterum modicum, et videbitis me: quia vado ad Patrem. 17 Dixerunt ergo ex discipulis ejus ad invicem: Quid est hoc quod dicit nobis: Modicum, et non videbitis me; et iterum modicum, et videbitis me, et quia vado ad Patrem? 18 Dicebant ergo: Quid est hoc quod dicit: Modicum? nescimus quid loquitur. 19 Cognovit autem Jesus, quia volebant eum interrogare, et dixit eis: De hoc quæritis inter vos quia dixi: Modicum, et non videbitis me; et iterum modicum, et videbitis me. 20 Amen, amen dico vobis: quia plorabitis, et flebitis vos, mundus autem gaudebit; vos autem contristabimini, sed tristitia vestra vertetur in gaudium. 21 Mulier cum parit, tristitiam habet, quia venit hora ejus; cum autem pepererit puerum, jam non meminit pressuræ propter gaudium, quia natus est homo in mundum. 22 Et vos igitur nunc quidem tristitiam habetis, iterum autem videbo vos, et gaudebit cor vestrum: et gaudium vestrum nemo tollet a vobis. 23 Et in illo die me non rogabitis quidquam. Amen, amen dico vobis: si quid petieritis Patrem in nomine meo, dabit vobis. 24 Usque modo non petistis quidquam in nomine meo: petite, et accipietis, ut gaudium vestrum sit plenum.25 Hæc in proverbiis locutus sum vobis. Venit hora cum jam non in proverbiis loquar vobis, sed palam de Patre annuntiabo vobis: 26 in illo die in nomine meo petetis: et non dico vobis quia ego rogabo Patrem de vobis: 27 ipse enim Pater amat vos, quia vos me amastis, et credidistis, quia ego a Deo exivi. 28 Exivi a Patre, et veni in mundum: iterum relinquo mundum, et vado ad Patrem. 29 Dicunt ei discipuli ejus: Ecce nunc palam loqueris, et proverbium nullum dicis: 30 nunc scimus quia scis omnia, et non opus est tibi ut quis te interroget: in hoc credimus quia a Deo existi. 31 Respondit eis Jesus: Modo creditis? 32 ecce venit hora, et jam venit, ut dispergamini unusquisque in propria, et me solum relinquatis: et non sum solus, quia Pater mecum est. 33 Hæc locutus sum vobis, ut in me pacem habeatis. In mundo pressuram habebitis: sed confidite, ego vici mundum.
And the Douay-Rheims:
A little while, and now you shall not see me; and again a little while, and you shall see me: because I go to the Father.  Then some of the disciples said one to another: What is this that he saith to us: A little while, and you shall not see me; and again a little while, and you shall see me, and, because I go to the Father?  They said therefore: What is this that he saith, A little while? we know not what he speaketh.  And Jesus knew that they had a mind to ask him; and he said to them: Of this do you inquire among yourselves, because I said: A little while, and you shall not see me; and again a little while, and you shall see me?  Amen, amen I say to you, that you shall lament and weep, but the world shall rejoice; and you shall be made sorrowful, but your sorrow shall be turned into joy.  A woman, when she is in labour, hath sorrow, because her hour is come; but when she hath brought forth the child, she remembereth no more the anguish, for joy that a man is born into the world.  So also you now indeed have sorrow; but I will see you again, and your heart shall rejoice; and your joy no man shall take from you.  And in that day you shall not ask me any thing. Amen, amen I say to you: if you ask the Father any thing in my name, he will give it you.  Hitherto you have not asked any thing in my name. Ask, and you shall receive; that your joy may be full.  These things I have spoken to you in proverbs. The hour cometh, when I will no more speak to you in proverbs, but will shew you plainly of the Father.  In that day you shall ask in my name; and I say not to you, that I will ask the Father for you:  For the Father himself loveth you, because you have loved me, and have believed that I came out from God.  I came forth from the Father, and am come into the world: again I leave the world, and I go to the Father.  His disciples say to him: Behold, now thou speakest plainly, and speakest no proverb.  Now we know that thou knowest all things, and thou needest not that any man should ask thee. By this we believe that thou camest forth from God.  Jesus answered them: Do you now believe?  Behold, the hour cometh, and it is now come, that you shall be scattered every man to his own, and shall leave me alone; and yet I am not alone, because the Father is with me.  These things I have spoken to you, that in me you may have peace. In the world you shall have distress: but have confidence, I have overcome the world.
The opening verses of this section are an important lesson the necessity of the Cross, both in Christ's sacrifice, and in our own lives, as the commentaries contained in the Catena Aurea make clear:
CHRYS. But then, if one examines, these are words of consolation: Because I go to the Father. For they show that His death was only a translation; and more consolation follows: And again, a little while, and you shall see Me: an intimation this that He would return and, after a short separation, come and live with them for ever...
AUG. Which must be understood thus: viz. that the disciples sorrowed at their Lord's death, and then immediately rejoiced at His resurrection. The world (i.e. the enemies of Christ, who put Him to death) rejoiced just when the disciples sorrowed, i.e. at His death: You shall weep and lament, but the world shall rejoice; and you shall be sorrowful, but your sorrow shall be turned into joy.
ALCUIN. But this speech of our Lord's is applicable to all believers who strive through present tears and afflictions to attain to the joys eternal. While the righteous weep, the world rejoices; for having no hope of the joys to come, all its delight is in the present.
CHRYS. Then He shows that sorrow brings forth joy, short sorrow infinite joy, by an example from nature: A woman when she is in travail has sorrow, because her hour is come; but as soon as she is delivered of the child, she remembers no more the anguish, for joy that a man is born into the world.
AUG This comparison does not seem difficult to understand. It was one which lay near at hand, and He Himself immediately shows its application. And you now therefore have sorrow; but I will see you again, and your heart shall rejoice. The bringing forth is compared to sorrow, the birth to joy, which is especially true in the birth of a boy. And your joy no man takes from you: their joy is Christ. This agrees with what the Apostle said, Christ being risen from the dead dies no more (Romans 6:9).
CHRYS. By this example He also intimates that He loosens the chains of death, and creates men anew. He does not say however that she should not have tribulation, but that she should not remember it; so great is the joy which follows. And so is it with the saints. He said not that a boy is born, but that a man, a tacit allusion to His own resurrection.
Through the Cross comes the promise of responses to our prayers from the Father. But it is not an unconstrained promise:
AUG. The word whatsoever must not be understood to mean anything, but something which with reference to obtaining the life of blessedness is not nothing. That is not sought in the Savior's name, which is sought to the hindering of our salvation; for by in My name must be understood not the mere sound of the letters or syllables, but that which is rightly and truly signified by that sound. He who holds any notion concerning Christ, which should not be held of the only Son of God, does not ask in His name.
But he who thinks rightly of Him, asks in His name, and receives what he asks, if it be not against his eternal salvation; he receives when it is right he should receive; for some things are only denied at present in order to be granted at a more suitable time. Again, the words, He will give it you, only comprehend those benefits which properly appertain to the persons who ask.
All saints are heard for themselves, but not for all; for it is not will give simply, but will give you; what follows, Hitherto have you asked nothing in My name, may be understood in two ways: either that they had not asked in His name, because they had not known it as it ought to be known; or, you have asked nothing, because with reference to obtaining the thing you ought to ask for, what you have asked for is to be counted nothing.
That therefore they may ask in His name not for what is nothing, but for the fullness of joy, He adds, Ask and you shall receive, that your joy may be full. This full joy is not carnal, but spiritual joy; and it will be full, when it is so great that nothing can be added to it.
AUG. And this is that full joy, than which nothing can be greater, viz. to enjoy God, the Trinity, in the image of Whom we are made.
AUG, Whatsoever then is asked, which appertains to the getting this joy, this must be asked in the name of Christ. For His saints that persevere in asking for it, He will never in His divine mercy disappoint. But whatever is asked beside this is nothing, i.e. not absolutely nothing, but nothing in comparison with so great a thing as this. It follows: These things have I spoken to you in proverbs; but the time comes when I shall no more speak to you in proverbs, but I shall show you plainly of the Father.
The hour of which He speaks may be understood of the future life, when we shall see Him, as the Apostle said, face to face, and, These things have I spoken to you in proverbs, of that which the Apostle said, Now we see as in a glass darkly (1 Cor 13:12). But I will show you that the Father shall be seen through the Son; For no man knows the Father save the Son, and he to whom the Son shall reveal Him (Matt 11:17).