Minerals are really important early game and can be a big bottleneck to your expansion. So build mining stations on any that you find in your early systems.
I always build a second science ship pretty quickly and start looking for mineral rich systems and potentially colonizable planets.You will always get 2 other nearby colonizable planets in addition to your home system. Often they already are in your controlled space.
You can separate your starting fleet into 1 ship task forces, set them to evasive and then just go and recon in expanding circles for habitable planets. You might lose one though if you enter a system and are too close to hostiles but in general you can scout out a large area around your home system to help prioritize where you send your science ships.
Use the influence to build outposts to stake claims on areas that you don’t want your neighbors getting to first.
You can also spend them on edicts to help you out in specific science areas if your research points have become unbalance.
Make sure you keep an eye on your researchers ages and when they will reach then end of their lifespans. You want to have influence points available to replace them.
Make sure you have a decent home fleet 7+ corvettes by the time the age of piracy kicks off…usually in 2205 or so depending on when you get your first colony out. Not sure what the trigger is but you will want to be able to respond quickly so you don’t lose a mining station or two. There are also some early game events that can trigger which require you have a decent home fleet.
If you get an advanced ai start neighbor or a xenophobic militarist neighbor prepare for war early.You might not be interested in war at that stage but it will be interested in you.
The more planets and pops you have the more your technology cards will cost so you need to balance expansion.
Colonizing a planet costs a lot of energy so make sure you have 250+ stored up if you don’t have a very large monthly income.
Don’t be afraid to keep playing if disaster strikes. I’ve pulled myself out of some deep holes. Also done some stupid things and salvaged the situation with luck and miracles as well as crashed and burned. It’s all part of the fun.