Rev 21:27 says that nothing unclean shall enter Heaven, which is why it is necessary that there be a place where we are purified before we enter Heaven. 2 Maccabees explains that there must be a place, as there would be no sense in offering prayers for the dead otherwise. The Catechism, in section 1030ff, explains much about the doctrine of Purgatory. In Matt 12:31f, Jesus states that some things may be forgiven in the age to come, again providing evidence of forgiveness in the next life. Again, in Matt 5:23-26 and 18:18-35, Jesus tells us that we will never get out of our debt of sin until we have paid the last penny, implying that this place is not Hell, which no one will ever leave.
Aquinas tells us that the purgatorial fire is a purifying, rather than afflictive fire. Mortal sin deprives us of communion with God, and this deprivation is eternal punishment in Hell. The suffering in Purgatory, on the other hand, is that of yearning for God, and it is not God's vengeance, and the Church has always commended almsgiving and prayers for those who are in Purgatory.
The Limbo of the unbaptized infants is something that scholars and saints have had myriad opinions on, with a variety of beliefs such as (1) unbaptized infants go to Hell, (2) that the infants go to Hell, but do not have the torments of Hell, (3) that they live on the edge of Heaven, a sort of limbo, or (4) that God will save unbaptized infants, and we should trust them to God's grace. It is not known for certain what the whole truth is, but even those who postulated Limbo believe that parents may save their children by desiring baptism for them.