The Bible teaches
...(1) that some sins are forgiven in the next world; (2) that some souls are saved in the next world "by fire"; (3) that it is useful and beneficial to pray for the dead; (4) there will be degrees of punishment and reward.
Sins forgiven in the next world.
Matt. 5:25-26 Be at agreement with thy adversary betimes, whilst thou art in the way with him: lest perhaps the adversary deliver thee to the judge, and the judge deliver thee to the officer, and thou be cast into prison. 26 Amen I say to thee, thou shalt not go out from thence till thou repay the last farthing.

Matt. 12:32: And whosoever speaketh a word against the Son of Man it shall be forgiven him: but whosoever speaketh against the Holy Ghost, it shall not be forgiven him, neither in this world, NEITHER IN THE WORLD TO COME.
(Some sins can therefore be forgiven after death.)
Saint Augustine (De Civ. l. xxi. c. 13.) and Saint Gregory (Dial. iv, c. 39.) gather, that some sins may be remitted in the world to come; and consequently that there is a purgatory, or a middle place. Saint Augustine says thse words would not be true, if some sins were not forgiven in the world to come; and Saint Gregory says, we are to believe from these words in the existence of the fire of purgatory, to expiate our smaller offences, before the day of judgment. Saints Isidore and Venerbale Bede say the same.
Saint Bernard, speaking of heretics, says, they do not believe in purgatory: let them then inquire of our Saviour, what he meant by these words.

Matt. 18:23-35 Therefore is the kingdom of heaven likened to a king, who would take an account of his servants. 24 And when he had begun to take the account, one was brought to him, that owed him ten thousand talents. 25 And as he had not wherewith to pay it, his lord commanded that he should be sold, and his wife and children and all that he had, and payment to be made. 26 But that servant falling down, besought him, saying: Have patience with me, and I will pay thee all. 27 And the lord of that servant being moved with pity, let him go and forgave him the debt. 28 But when that servant was gone out, he found one of his fellow servants that owed him an hundred pence: and laying hold of him, throttled him, saying: Pay what thou owest. 29 And his fellow servant falling down, besought him, saying: Have patience with me, and I will pay thee all. 30 And he would not: but went and cast him into prison, till he paid the debt. 31 Now his fellow servants seeing what was done, were very much grieved, and they came and told their lord all that was done. 32 Then his lord called him; and said to him: Thou wicked servant, I forgave thee all the debt, because thou besoughtest me: 33 Shouldst not thou then have had compassion also on thy fellow servant, even as I had compassion on thee? 34 And his lord being angry, delivered him to the torturers until he paid all the debt. 35 So also shall my heavenly Father do to you, if you forgive not every one his brother from your hearts.

Luke 12:47-48 And that servant who knew the will of his lord, and prepared not himself, and did not according to his will, shall be beaten with many stripes. 48 But he that knew not, and did things worthy of stripes, shall be beaten with few stripes. And unto whomsoever much is given, of him much shall be required: and to whom they have committed much, of him they will demand the more.

Apoc. 21:27 There shall not enter into it any thing defiled, or that worketh abomination or maketh a lie, but they that are written in the book of life of the Lamb.

Some souls are saved in the next world "by fire".
I Cor. 3: 13 and 15: Every man's work shall be made manifest: for the day shall declare it, because it shall be revealed by fire and the fire shall try every man's work of what sort it is. 15. If any man's work shall be burned, he shall suffer loss; but he himself shall be saved, YET SO AS BY FIRE.

It is useful and beneficial to pray for the dead.
2 Machabees 12:46 It is therefore a holy and wholesome thought to pray for the dead that they may be loosed from their sins.

Ecclus. 7:32 A gift hath grace in the sight of all the living, and restrain not grace from the dead. (This is one of the Old Testament books omitted from the Protestant Bible - even the Jews accept this one)
There will be degrees of punishment and reward.
Rom 14:22 Every one of us will render an account for himself to God.
Matt. 12:36 Of every idle word men speak, they shall give account on the day of judgment.
Luke 12:48 but of every one to whom much has been given, much will be required; and of him to whom they have entrusted much, they will demand the more.