Slay the Spire 1 always runs the same acts in the same order. Slay the Spire 2 has two possible first acts and one fixed act each for acts 2 and 3.

StS1 acts

ActNameBosses
1ExordiumThe Guardian, Hexaghost, Slime Boss
2The CityThe Champ, Bronze Automaton, The Collector
3The BeyondAwakened One, Time Eater, Donu and Deca
4The EndingCorrupt Heart, with Spire Shield and Spire Spear as the elite

Act 4 opens only when you hold all three keys (Ruby, Emerald and Sapphire) after beating the act 3 boss.

StS2 acts

ActOptionsRoomsWeak fights at the startRest sitesBosses
1Overgrowth1536 or 7Vantom, Ceremonial Beast, The Kin
1Underdocks1536 or 7Waterfall Giant, Soul Fysh, Lagavulin Matriarch
2Hive1426 or 7The Insatiable, Knowledge Demon, Kaiser Crab
3Glory1325 or 6Queen, Test Subject, Aeonglass

Each act adds two floors to its room count: the Ancient floor at the start and the boss floor at the end. In multiplayer every act has one room fewer.

Overgrowth and Underdocks get 7 rest sites about 61% of the time and 6 otherwise. Hive gets 6 about 61% of the time and 7 otherwise. Glory gets 5 or 6 with equal chance.

Which act 1 you get

Underdocks is locked at first. It unlocks from score accumulated across runs. In the first single-player run after it unlocks, act 1 is always Underdocks. After that, act 1 is picked at random between Overgrowth and Underdocks.

Which boss you meet

Each act lists its bosses in a fixed order. In a standard run the act uses the first boss in that order that you have never fought. Once you have seen all three, the boss is random. StS1 uses the same rule for its first three acts.

At Ascension 10 in StS2, act 3 adds a second boss, drawn from the two act 3 bosses you are not already fighting.

Ancients per act

ActAncients that can appear
Act 1Neow
Act 2Orobas, Pael, Tezcatara
Act 3Nonupeipe, Tanx, Vakuu

Darv is shared: once unlocked, he can appear in act 2 or act 3. See Neow and the Ancients for what each one offers.