The most basic truth of 2.2 is population is most important resource, so you should colonize early and prioritize getting planets over anything else.
That said, empire sprawl is annoying. Do your best to avoid star systems that have few resources or don´t lead to more planets. Make sure you dont break empire cohesion though. If you see empire cohesion affecting your empire sprawl, you will need to fill in a few holes. All planets must have a path leading back to capital for trade purposes, so make sure your trade routes aren´t flowing through unclaimed space.
As to what to do at start of game? Science ships. Should get like 5 of them surveying asap. The abnormality bonuses are so vital early game, you will want to take map the stars edict first thing.
As to what do to with your planets, well you should turn on resettlement if not already allowed, and try to grow your colonies to 10 pops asap to upgrade colony to planetary admin and get 1 buildings on them, this building is temple for spiritualists, or robot factory otherwise. This is mostly to remove the -50% growth modifier. To that end, you should probably only build a second alloy factory and temple/robot plant on homeworld, then you will be able to use more homeworld pop on colonies without having to lose full building slots.
Once you have a few pop 10 colonies, you can use their population to feed other colonies to size 10. Fun fact, planetary admins dont degrade if pop goes under 10, but don´t go under 5, or you will lose that 1 building.
(edit) Yes, it is possible to have breeder planets, that do nothing but produce robots (or unity) and population, and have nothing but 1 city district, a robot factory (or temple), and a plantary admin. As soon as an unemployment icon shows up next to a breeder planet, resettle that pop elsewhere. Many players will create breeder planets with yellow or even red habitability worlds. This strategy works... but you will need to really produce a lot of consumer goods to sustain it.
Always try to keep the Encourage Planetary Growth decision on all worlds, so you will need more farms than any other resource type, if your average population is under 20 per planet and you have tons of food, you should go ahead and use nutricional plentitude policies too. Since that only happens in early game, you should probably turn it on at about time you have 2 colonies with planetary admins, and turn it off later when you get to 20+ pop per planet or start running low on food production. Be careful you will be locked into it for 10 years.
You should also probably turn on consumer benefits trade policy, as it will reduce the number of civilian factories you need, which is helpful since the defualt setting ¨militarized economy¨ gives you +15% alloy production and -25% consumer goods production, but that doesn´t apply to your consumer goods produced by trade.
If you take the thrifty race perk and 1 point into authoritarian ethics, you should consider moving excess population back to homeworld and building commerce centers as a way to produce consumer goods instead of civilian factories. Also prosperity tree perk that gives 1 extra clerk job per city disrict is a must have for this type of clerk build.
Otherwise watch your alloy factories to civilian factories ratio and remember to switch to mixed economy policy if that ratio falls under 2 to 1.