General:
* Follow
* Obey
* Comply with
* Conform to
* Stick to
* Abide by
* Keep to
* Observe
* Uphold
* Respect
Formal:
* Heed
* Execute
* Implement
* Enforce
Informal:
* Go by
* Live by
* Hold to
More specific:
* Embrace (an idea)
* Stand by (a principle)
* Maintain (a position)
* Commit to (a plan)
The best synonym for "adhere to" will depend on the specific context of your sentence. For example, if you're talking about following rules, "obey" or "comply with" might be more appropriate. If you're talking about sticking to a plan, "keep to" or "stick to" might be better choices.
Consider the tone of your writing and the specific meaning you want to convey when choosing a synonym.