General:
* Found
* Create
* Set up
* Institute
* Inaugurate
* Start
* Begin
* Form
* Build
* Erect
* Develop
More specific:
* Confirm (to establish something as true or certain)
* Prove (to establish something with evidence)
* Secure (to establish something firmly in place)
* Determine (to establish something with certainty)
* Define (to establish the meaning of something)
* Fix (to establish something firmly)
* Settle (to establish something permanently)
Figurative:
* Plant (to establish something firmly)
* Anchor (to establish something securely)
* Ground (to establish something firmly)
The best synonym will depend on the specific context of your sentence.