Hello.. How do I establish a home planet? thanks

You need to claim a base on a planet of interest. There are many youtube videos to watch or the quick summary is:

Scan from space and look for a habitable base. For your first base there was (or perhaps still is) a bug where you would need to do this on a plant without visible liquid/water ponds or lakes.
Fly to base location and land.
Enter the single-room base and there is a red and silver tall console with a computer screen (usually to the left) and mousing over will bring up a pop-up containing words to the effect that lists the base name and "interact//habitable base."
Hold "E" (or whatever the controller key is) and follow the prompt to claim.

You now are the proud owner of a base. You can only have one active base at a time and will also then have to go through the steps to build the terminals then find and hire the "specialists" to obtain blueprints as well as expanding your base. These are typically Fed-Ex type "quests."

Caution - moving your base without first tearing it apart will mean large losses in resources as only half the resources will transfer and you would then have to search for and craft replacement resources.

Some suggestion to add to what Chewy has said:

When you enter a new system, do a scan from your spaceship. Not every system will respond to a scan with a habitable base but most will.

There may be (will always be?) habitable bases that don't show up with the "red hexagon" tag. If you're flying around a planet and notice a round meadow with a single circular building in the center of it, that's likely to be a base which you could claim.

As Chewy says, moving your base is expensive so choose with care. There are bases on planets with extreme weather and extreme sentinels -- if you really want to punish yourself. There are also bases on planets with mild weather, decent resources, and no or few pesky animals trying to chew you up for dinner. You're going to stick with that base for a while so might as well pick one with a nice view where you don't use up resources keeping your suit shield fed when you venture outside.

Don't go nuts with building a fancy base. The "normal" mode has limits on how complex the base can be so you may find yourself having to disassemble and rebuild parts of it in order to make room for, say, an Albumin Pearl farm.

You can disassemble all base elements [*] without losing resources, so you're not stuck with your first layout. Just do have room in your exosuit for the resources. If you disassemble a storage unit, the stored contents don't show up in your suit inventory but they will magically reappear when you rebuild that storage box.

And speaking of disassembling, after you finish with the various specialists at the terminals (farmer, scientist, etc.) you can disassemble their terminals and recover the resources. The NPCs disappear with the terminal but will reappear if you decide to build that terminal type again.


[*] You can construct some things away from your base, like save points and beacons. AFAIK, when you're not on your home planet then deconstructing those items only returns half of the resources.