Looking at the text of the bill itself[^1], there is no date of effectiveness, so it would take effect immediately upon signing. States currently not observing DST (i.e. Hawaii and most of Arizona) can remain on standard time, but everyone else would be advanced an hour year-round. Because no territories currently observe DST, the bill would de facto have no effect there.
Interestingly, there is nothing that specifies whether standard or daylight time is used in Hawaii, Arizona, and the territories if they don't explicitly pass a bill, but I presume common sense would prevail in that nothing changes.
A parallel bill (S.29) was introduced in the Senate.