"If I create an LLC in Ohio, what types of fees are involved?"
You have to file the LLC's Articles of Organization with the Ohio Secretary of State. The annual fee as of today is $99.
Then of course you'd have to draft an LLC's operating agreement. You could hire an attorney to do this for you or you could grab some boilerplate template online that may or may not be worthless. This fee of course varies depending on the approach you take.
You also need a registered agent in Ohio, though there are many services who will serve as your statutory agent (equivalent of a registered agent in California) for a nominal annual fee.
"Do I need to pay CA their minimum $800/year LLC fee if I register it as an Ohio LLC....?"
The answer is almost always YES since California considers an LLC as "doing business in California" and therefore required to register with the California Secretary of State, file the annual Form 568, and pay the annual $800 LLC tax if you, as a member of the LLC, do pretty much anything anything on the LLC's behalf while you are in California.
Below are two excerpts from FTB Form 3556 that make the state's position pretty clear.
"Thoughts and advice are greatly appreciated!"
Consult with qualified legal and tax advisors before forming an LLC.