In the first stanza, he is contemplating the vase in its entirety. He marvels at the piece's perfection (still unravish'd bride of quietness) despite its age (foster child of Silence and slow Time). He then feigns ignorance of Greek culture to reinforce this, asking "What men or gods are these?" etc. Basically he is making the point that so much time has passed that whole cultures have vanished yet this urn has survived unscathed. If you really think about the 2600 years that have passed since pots of this type were made in Ancient Greece, and all the people who have lived and died since then, and all the wars that have been fought and all the destruction that has ravaged the world, then you will begin to see what astonished Keats. That despite all of that something so fragile and detailed and beautiful produced by a skilled hand remained unspoiled and beautiful. This is the sense of awe with which the poem must be understood.
In the second stanza, he is speaking to the individual painted figures. At first to a lute player. Though the lute player is not real and cannot really play (being a painted figure), Keats envies him the eternity which he will spend lost in his song (most educated people could play some sort of instruments back when Keats lived so he and his readers would have known the joy of being lost within music while playing it). The lute player is evidently playing beneath some trees which Keats says will never lose their leaves. Again, he is envying the fact that the figure will get to enjoy an eternal summer. He then speaks to another figure. A young man trying