I would say
A week today
Three weeks today
but looking on GloWbE, it's pretty clear that these are British (and Commonwealth) English. In American English, the equivalent is
A week from today
Three weeks from today.
(I would never say these).
In British English we can also say
A week next Tuesday
and even
Tuesday week
but I don't think Americans say these much.
I'm not sure about
A year today
but I think it works - certainly there are a few instances of this in GloWbE.
So to answer your question directly:
Three weeks today (BrE)
Three weeks from today (AmE)
In three weeks' time (both).