Start a game: Single Player -> Quick Start -> Enter a name for your campaign at the bottom -> Scenario 1 - Classic Fleet - Difficulty 4. Pick scenario 2 or 3 if you're good at strategy games or were able to beat AI War 1 on a similar difficulty.
To win the game you must find the AI homeworld and destroy its base. Everytime you capture a new planet, more of the map is automatically revealed.
You start with some fleets. Fleets autobuild units as long as there is a factory or mobile factory on the same planet. Flagships can be crippled but never killed.
Tab opens the galaxy map. You can select and move your fleets and individual units from here or from the planet view.
L key loads a fleet into its flagship, U key unloads it. Transports are tough, use that to reach distant targets.
Ctrl+right-click on a wormhole to send units through, or alternatively right-click the destination, whether it be a specific location or a planet on the galaxy map.
Capture a planet by destroying the enemy command station and building your own (Build tab). You can often manage to do this while there are still enemies in the system! (Note that your units won't target automatically the command station, warp gate or other building which increases AI Progress, you need to right-click it yourself.)
Building units costs metal per second, and how many units (and turrets) you can have is capped by energy. Captured planets auto-build metal harvesters, yields vary. Logistical Command Station is often a good pick to build as it gives you a good amount of energy while still letting you build decent defences.
Any AI-controlled neighbouring planet with a Warp Gate building can send periodic attack waves against your planets. Build turrets (Build tab) to stop those, or do a sneaky raid to kill the gates in the directions you don't want to get waves from. (You can concentrate all waves on a single planet this way.)
Capture stuff like new fleets, Global Command Augmenters and Coordinators to get more powerful. Look at the Local and Hacking tabs from the galaxy map for every neighbouring world and read all the building descriptions to see what's interesting.
Destroying a command station or warp gate angers the AI, a.k.a. increases AI Progress (AIP). Capture enough territory to support your economy, acquire weapons and have safe planets for economic stations, but don't try to capture the whole map like your typical Civ or space 4X game.
Try to ignore or neuter (destroy everything except command station) worlds that don't offer you much. Hacking lets you often get stuff without having to destroy the AI command station.
Capturing planets is the main source of science points (to upgrade stuff) and hacking points. Just having a command station there starts harvesting them, it's a valid strategy to strip the science and hacking points from a planet and then abandon it - one less thing to defend.