In general, to win you must destroy the Fort and Keep within the same single lane and then Core. You can backdoor with certain Heroes (notably Sgt. Hammer) as the walls/gates/Towers need not be destroyed.
The exception is Towers of Doom, where capturing and then defending all 6 Forts deals structure damage every ~3.5 seconds and also increases the damage dealt by channeling each Altar to a maximum of 7 and is therefore also a fast, but not the only, way of winning.
The gates open when the timer hits 00:00 (no negative sign) on every map. You should test to see the fastest way to get to a particular lane if you're going to cheese towers a bit before minion waves collide. Mid might be faster than the gate directly in the lane you want to push.
Remember that both your Health and Mana are resources. If either is totally full, you may not be using your resources efficiently. Therefore, most Heroes should run into 1 tower at the start of the game and do a little damage, especially if they are ranged and can at least do a little harm, to the point that they take at least 1 Tower shot (stand to the side so you don't tank both Towers). Most Heroes regenerate about 12.5% of their max Health every minute, and most Heroes have 490 mana + 10 per level as well as regenerating 0.9 + .1/level mana/s.
Minion XP globes are dropped by all Minions except Reavers/Catapults/Leader (Wizard) and give 80 XP + 2 per minute. Fort Towers give 125 XP (except 250 on Towers of Doom) while Keep Towers give 400 XP. You passively gain 23 XP per second starting 35 seconds into the match, increased by 20% stacking additively for each Fort or Keep destroyed, except Towers of Doom where it gives a large flat XP bonus upon capture.
Each level increases most Heroes' health and damage (both auto-attacks and abilities) by 4% per level, stacking multiplicatively. Therefore, it may be to your advantage to prioritize soaking early, rotating between lanes to get all minion XP, or splitting with Vikings, rather than only hard shoving 1 lane, on certain maps, especially to hit the lvl 4 power spike (Mercenary Lord talent for multiple Heroes).
Minions do triple damage to enemy Structures, although they attempt to focus enemy Minions over Structures.
Mercenary Camps spawn at 1 minute on every map, except Boss mercenaries which spawn at 5 minutes. Mercenary respawn timers are too long (at least 2:30) to be worth worrying about for most speedruns, except the Gnoll siege camp on Alterac which is on a short 1:30 timer.
Siege Giant camps do splash damage and have 10 attack range (higher than Structures which have at most 8) and therefore can be kept alive and pushed with indefinitely. When capturing them, their attacks are telegraphed with a red circle in advance and they therefore can be kited to take 0 damage from them while attacking at max efficiency.
The Raven from the Bruiser camp on Starcraft maps has 11 range, although its damage is low.
The Assault Troopers on Volskaya also have 10 range, but they don't do splash damage and do less DPS than the Giants.
The Hellbats from siege camp on Starcraft maps (and Gnoll son Alterac, which are reskinned Hellbats) apply an armor debuff, even to enemy Structures, of up to -20, increasing damage dealt to them by 20%.
Map specific tricks:
Braxis Holdout and Dragon Shire objectives both start at 1:30. Channeling each Beacon/Shrine takes 4 seconds. On Dragon Shire, channeling the Knight takes 3 seconds in mid lane.
On Braxis Holdout, you should be able to end with the first Zerg Wave with any Hero. Note that the first wave's lane is randomly selected (subsequent waves alternate sides), and is not revealed until the 1 minute mark. So if you're going for a record, you may need multiple tries just to make sure your time in the first minute was not wasted. The only way to know in advance which side will be pushedwould be to reverse-engineer how the game chooses a random seed for its Rand() function, maybe it just uses the UNIX time stamp at game start on the server?
On Dragon Shire and Volskaya, the Vehicle's strength scales and is determined by the minute mark when it is captured.
On Blackheart's Bay, you can start turning in coins at 1:15, and the first chests spawn at 1:30, giving a total of 10 coins just from the chests, enough for turn in; however, the first turn in only attacks mid, which is a bad lane to push as it has no camps.
On Garden of Terror the first objective is at 2:30.
On other maps the first objective is at 3 minutes.
On Volskaya Foundry, the first Objective is always mid, and capturing the Objective requires standing on the point for 6 seconds, then an additional 52 seconds for it to hit 100%. Once the timer starts ticking up, you can leave for up to 6 seconds at a time while it continues ticking up (to push mid, or soak and therefore level up).
TBD: Hero specific tricks (especially Sylvanas, Vikings, Hammer)
