A technique I find helpful when dealing with Paul's long sentences: break up the phrases in terms of conjunctions and prepositions. Once you do that, and especially if you're willing to use some spacing to further clarify, Paul's flow of thought becomes a lot clearer. That, in turn, lets us reason carefully through what he is saying, and apply it to our own lives more fruitfully.
But when the goodness and loving kindness of God our Savior appeared, he saved us
not because of works done by us in righteousness
but according to his own mercy
by the washing and regeneration of the Holy Spirit
whom he poured out richly on us through Jesus Christ our Savior
so that (being justified by his grace) we might become heirs
according to the hope of eternal life.
This helps us see why God saved us (not because... but according to...), how he saved us (by...) and a little about that process (whom...), and to what end (so that...). As a single block, that's hard to parse and follow, but with each clause broken up and its subclauses distinguished visually, it's much easier to understand. This is pretty much what I do whenever I find a tangled verse—whether it's in Paul's writings, or anywhere else. Perhaps it will help you a bit along the way as well.