Ty Ara.
My initial game was with the military human, influence there was hard, also because it was my first game. The second human race, the democratic one, gets an influence of influence every four years, with the presidential mandate. So by just expanding and building I fulfill the mandate and boom easy influence.
For early expansion I'm having more success with frontier expanding than with planet expansion. I disable the inner frontiers after I have expanded my borders a second time.
I read you barely need frontiers, but I'm having problem not relying on them.
What is a good colonization strategy?
I used to have influence problems all the time until I realized that you will never gain more than 5 influence points per turn even at your best, so you better ration it wisely. So I tend to use influence only as much as needed and typically thats recruiting leaders, colonizing planets and sometimes I will have 1 empire edicts running (one of the tech grants that gives a 20% bonus for research)
I play a 5 Core strategy which basically is a rush to colonize 5 planets as fast as possible, build up your fleet to keep enemies at bay and set yourself up for the mid game.
I break up my first 3 Corvettes into separate fleets and have them scout all the star systems around me for planets. Typically I can scout 20-30 star systems in first 2-3 months. At the same time I have my science ship survey home system, while I immediately start building 2nd science ship on 1st turn. By now 1st ship finishes surveying home system, 2nd ship finishes building - now I can send them directly to star systems where corvettes found planets and see whats habitable. This saves you a ton of time from blindly surveying nearby systems. Corvettes keep fanning out and searching more systems for planets. At this point you probably have 500 minerals, start building first colony ship. Only make improvements on homeworld to keep unemployed pops working (could be minerals, farms or energy whatever needed, but no research buildings). You want to maximize minerals yield right now so you can crank out your 5 colony ships and start applying surpluses to building planetary buildings, warships, making upgrades, etc. Also, dont build any mining or research stations in space, you dont break even typically for 3+ years at this point so you're not in position to make those investments yet.
From the beginning I adopt Traditions in this order to minimize credit expenditure and maximize colonization and growth speed, this will probably take a few years
Expansion adoption
+100% Colony development speed
Supremacy adoption
+20% Border range
Harmony adoption
+25% Growth speed
After a year, your first colony ship is finished building. You probably found 2-3 habitable planets by now, pick what you think is best one, either by size or habitability. The further away the better to check the expansion of other empires. Go colonize it, start building another colony ship right away. You should be making enough minerals (500+ a year) at this point to keep colony ships constantly producing 1 at a time until you make 5 and colonize 5 planets. In the mean time, research tech that will help you maximize planetary yields so Mine II, Farm II, Power Plant II, etc. Gradually build up your military fleet, keep regularly checking the diplomacy screen to check the relative strength of your opponents power (see screenshot below) to see if they are outpacing you in military strength and offer everyone a non aggression pact to get some breathing room. You can also guarantee independence of states that are further away from others so they are less likely to get into an early war, that helps you get your diplomacy score up early with them and set up a non aggression pact and research agreements. Research agreements are also really important as they can boost your tech research 25-50%. Discontinue independence guarantees as you get closer to mid game so you dont get sucked into someone else's war. I also trade star charts with friendly states to minimize exploration/surveying time to find habitable planets. Keep doing all the quests in the mean time too, its an easy way to pick up minerals, energy, influence, etc.
So just keep doing this, now you'll start seeing some energy and mineral surpluses. Start looking around for high yield space resources (4 minerals or 4 energy) to see where you can go send your constructor ship and build stations over them.
Basically thats what I do. Then I typically just build up during mid game, wait for a war to break out around me, so I can join in and attack whos likely going to lose and pick off a bunch of planets for myself. Then beat up on smaller empires around me and take more planets, then get ready for the final slugfest with the final empire whos probably same size as me by the point.