Besides the mining bases you seem to have figured out, there are three other options (four in TA, but terror stars are a joke and you don't have access anyway). All can be started by moving a constructor to where you want to place the base, then clicking the "construct" option in the bottom center window (where you see ship and/or planet information on what you have selected). This will bring up the window where you choose which type of base you want to build: military, economy, or influence. All of these can be upgraded by running another constructor into them, just like mining bases.

Military bases get modules to assist nearby ships with attack, defense, and speed. Keep in mind you need at least one point in attack and defense to get any benefit. A one point laser ship will get all weapon bonuses, but not defense bonuses. A one point armor ship will get all defense bonuses, but no attack. A ship with both defense and weapons will get all bonuses.

Economy bases can boost research and manufacturing on planets inside their area of effect. They can also boost the income from your freighters which pass through the area.

Influence bases do NOT boost the influence of planets in their ring - they generate their own influence, and the modules added to them boost the base influence they generate. Also, ignore the area of effect of these bases, the effect tapers off with distance, so for best effect place them touching the planets you want to influence.

All bases get weapons and defenses to help defend themselves, but even with all techs available they cannot defend themselves against a real attack. It's often not worth the time and money to defend bases with modules, use ships instead.