Landlord laws in Ohio can be found here:
http://codes.ohio.gov/orc/5321
Ohio is a pretty Landlord friendly state so there's not a ton of laws/rules. There may be city ordinances that you have to follow so you might want to check on that.
I guess I don't know what a "triplex" is, but if they are all separate units you don't need to notify other tenants that you will be in another unit, you only have to notify the occupant(s) of that unit, and you have to give 24 hour notice unless it's an emergency. There are no laws that I am aware of about how notice is given. You want to be a good landlord and you want your tenants to feel secure so do your best to notify them, whether that's by phone or email or text or a note on the door or whatever, you want them to know you're coming and you preferably want them to acknowledge your notice. In your case though where the apartment is empty, no notice is required assuming your tenant has terminated tenancy officially.
You can mow grass whenever you feel like it. A nice landlord might want to let folks know that you'll be there "every Saturday around noon" or whatever, but it's not necessary.
The suggestion about a local landlord group if you can find one is a good one. A lease can be difficult, and sometimes local judges can rule differently on lease clauses so it's good to have a local reference if you can find one. If nothing else, this might be the time to spend a little money with an attorney and have them help you with the official paperwork.