Some quick tips from someone who sucked really badly to someone who seems to have gotten the hang of it.
1. Don't be afraid to play around with species choices/making a custom species - you might have a particular play style that is really favourable to particular traits.
2. Some resources are very important early on - I used to ignore food, but then I realized how absolutely important it was early on, you colonize a solar system and pump food upgrades into it, and then you have more people - and all those modifiers (+1 blank per person on planet) start to really add up quickly.
3. Take a good long look at the system and planetary modifiers. You think that upgrading your planet modifiers for example are silly, but they are a HUGE boost - especially considering they are retroactive.
4. Try your best to always have 2+ people per planet on a solar system, constantly making pilgrimage ships can cripple your growth, especially early on - it will slow you down.
5. Solar system upgrades are -key-, need to keep your population happy? There is a solar system upgrade for that. A few actually, and some are cheap.
Little tips people might not realise -
You can shift click and select specific tech upgrades in any order you want
You can 'pollinate' a solar system you're already in without building another pilgrimage ship - and it's cheaper (I didn't realize you could just click on another planet in the system, and when you have 2 or more people on a planet, can populate the new planet)
Don't go to war right away - put a few points in being peaceful and diplomatic trades, and hold off a war until your ready and comfortable
Any planet with a garden of Eden trait is gold
I looked for this specifically, coming from Civ, and I haven't noticed it either.
It's there, you can see it in your 'where is all my gold going' breakdown in your empire screen when you hover over your cash.