How many Dominion sets are there?
As of 2020, there are 14 sets and 11 promo cards1.

In release order:


Dominion (Base game)
Intrigue
Seaside
Alchemy
Prosperity
Cornucopia
Hinterlands
Dark Ages
Guilds
Adventures
Empires
Nocturne
Renaissance
Menagerie


And the list of Promo cards:


Envoy (November 2008)
Black Market (March 2009)
Stash (February 2010)
Walled Village (June 2011)
Governor (October 2011)
Prince (June 2014)
Summon (November 2015)
Sauna/Avanto (September 2016) 2
Dismantle (December 2017)
Church (2019)
Captain (2019)


Separately there is also the Dominion Base Cards product; which includes the treasures and victory cards you need to play any setup of Dominion; without including any Kingdom cards. This allows players to purchase just an expansion without needing the base game, and allows 5-6 player games. 

Guilds and Cornucopia, each being half-sized expansions, are currently only sold as a combined Guilds&amp;Cornucopia set.

Dominion Base Game and Intrigue each have both a first and second edition, with the second edition for each removing 6 cards and adding in a new 7 cards. Due to some cards being different; you could count first edition and second edition as separate sets. Seaside and later have received minor updates in later printings; but nothing as big as brand new cards. Upgrade packs including just the 7 new cards were released for each.

Note that all sets through Guilds were planned as part of the original design of Dominion; while Adventures and beyond were created later due to Dominion's popularity.

All information taken from http://wiki.dominionstrategy.com.

1 "Cards" is used here to refer to all promos including Summon, even though Summon is an Event, not a Card, within the rules of Dominion. 

2 Sauna/Avanto is a split pile; counted here as 1 card even though it is technically 2 separate cards.