1. During the first playthrough, choose the Imperials (i.e. humans - the United Nations), Sophon or a bit more challenging Horatio. All of the above colonize the same way. They need a colony ship, an empty system with habitable planets. Humans are dependant on the influence, which allows them to facilitate building or developing technologies. The Sophon are great scientists and provide additional bonuses if they're the first to discover something and know the surrounding systems. Horatio have the "gene hunter" faction affinity - they can import the genes of other races and merge them with their own. They also have a default bonus to happiness per capita and begin with the ecologist part ruling (they're able to colonize almost every planet with only a minor penalty to production).
2. Exploration - looking for habitable planets and investigating anomalies should be your priority during the initial turns. Look for habitable systems that are reach with resources. If you run out of options, send out probes to discover new constellations. More scout ships means more probes - from time to time, send out a fleet of 3 ships commanded by a hero.
3. Colonize the most relevant systems. Look for systems that have many habitable planets or that have a lot of resources. Some of these planets may be home to many anomalies, which may render colonization rather futile. Colonize planets closer to the capital, which will help them reach the "colony" status sooner.
4. Technologies. You will have to decide which inventions you find more useful pretty early in the game. If pirates and hostile factions are giving you a hard time - invest your time and resources into industry and weapon modules. Your civilization is small and/or isolated? Look into upgrading engines and accessing new methods of travel. Having hard time finding a habitable planet, or struggling with unhappy populations? Work on terraforming technology and improving happiness.
5. Consequent policies. This may be challenging, but the more often you opt for the same party, the more advanced rewards you'll have access to. Each party has their own progress bar. It really pays off to follow through a single option so make sure to sort the priorities at the beginning of the game, so that you will be able to realize them.
6. If you need a stronger fleet work on discovering news hip classes or improve the size of the fleet with command points. Installing better modules and hiring heroes can also help. If the enemy is blocking your system, it's sometimes wiser to hold fleets in hangars until you manage to assemble appropriate forces.
7. Conquer smaller factions close to your borders. This is the easiest way to expand your empire. Focus on military or diplomacy and you will be able to conquer/assimilate the given faction. If you're the first one to assimilate a faction, you will be able to control it and get an assimilation bonus for the whole empire.
8. Heroes. Make sure they're not idle. This gives them experience and each of them has some useful skills. Specialization is a good idea - example: assign only those skills that work inside a system (for governors) and improve the industry, or only those skills that impact fleets and improve space combat. The more they work the more XP they get. Their ships can be improved for a payment in Dust, which is expensive, but also allows to boost the ships instantly. Make sure you're making new agents - those are valuable assets.
9. Keep track of the races that inhabit your systems. There are many races that can provide additional income if some specific circumstances are met, such as if the planet is fertile or anomalies occur there. Some races require other races to function, such as the Craviers, who use other as their slaves - they become better workers, but a planet is slowly being depleted and the happiness plummets. You might sometimes want to move certain races to other systems, where they will be more productive. You have to raise the System Development parameter to be able to use the space port for transportation of passengers.
10. Strategic resources are imperative for production of ships and modules. Then, the luxury resources are used for system development, or increasing the number of citizens. They can also be used to bribe unhappy populations or just to trade with others. Both types of resources provide local bonuses for planets.