I'm currently doing similar. I like the BGS.
If you want to take over control of another factions base, you need to initiate a war or civil war. You do this by bringing your influence up to the same as the other faction.
If the faction is below your own influence then you need to bring them up to your level.
To increase yours or the other factions influence, you simply need to do missions that have 'influence boost' as the payment.
Everything you do to effect influence is calculated every 24 hours, currently at about 4PM each day.
elitebgs.app
A good site that I use to understand the BGS is this one. It's more up to date too.
novaforce.com
These are ok too.
Influence is a way for factions to have or gain control over a star system. Many factions can share influence within a system. The first row in the system's faction lists the controlling faction for a system and always controls the controlling station, which is usually the biggest in the system...
elite-dangerous.fandom.com
The Background Simulation is dynamic, complex and reflects the evolution of the galaxy with ever changing power and system influence of factions in response to player behaviour. Every Station, Surface Port and Settlement has its own demands and supply that change dynamically. The BGS does not...
elite-dangerous.fandom.com