I suspect this has been discussed elsewhere at some point, but I thought it might be fun to get an exhaustive list of ways to get a Province on turn 3. Here are some off the top of my head. Several can be permuted in obvious ways by switching cards in your hand and those that are drawn.

Coppersmith, 4 Coppers (or 3 Coppers and a Silver)

Baron, Estate, Silver, 2 Coppers

Mining Village (trashed for $2), 4 Coppers, and a Silver (or $3 Silver-equivalent) on deck

Envoy, Silver, 3 Coppers, and some luck or a generous opponent

A lucky Tribute and 4 Coppers

Smithy, Silver, 3 Coppers, and 3 more Coppers on deck

Council Room, 4 Coppers, and 4 more Coppers on deck (one bought with the $2 opening)

I'll leave one more as a puzzle: how can you do it with Harvest?