[verb た-form] + 後で [noun] + の + 後で 後で + [sentence]
後で means "after". After a verb it takes the た-form, and after a noun it takes の. The verb stays in the た-form even when the whole sentence is about the future.
On its own, with no verb or noun before it, 後で means "later". In this sense it is often written in kana, あとで.
Both put one action after another. てから stresses the order – first this, and only then that – and it attaches to the て-form. 後で just places the second action somewhere later.
With で, the second half is one event that happens after the first. For a state that lasts from then on, drop the で, or use てから.
After a verb or noun, あとに後に means the same as 後で and sounds a little more formal.