I am a beef cattle farmer, from Ohio even, so I'm going to chime in here.
First off, THANKS Grrly for noting the difference between beef and dairy breeds. HUGE difference -- and chances are if you got yourself a butchered dairy cow, you'd hate it. Just personal experience -- less tender, less marbled, and therefore lots tougher. Think shoe leather.
As far as finding a farmer near you, that would probably be the best bet. You could check here:
Ohio Ecological Food and Farm Association Good Earth Guide:
Local Harvest (works nationwide)
I'm not sure what county you are in (Hamilton?) but another option is to call your county Farm Bureau office -- it is a farmer-run organization and no doubt they can direct you to someone pretty close to you (and someone who may not be listed at one of the above links). Plus, I always like to talk to a real person anyhow!
If you build a relationship with this person, you'll have your own personal farmer for life! (Provided he/she can stay in business.)
How it would work is that you call the farmer, say you're interested in buying 1/4 or 1/2 or a whole beef, and they'll tell you if they have anything ready and/or when they will have one ready. (It's ready when it's fat enough and big enough to give YOU a good product.) They'll deliver it to the slaughterhouse for you, and the butcher there will call you to see how you want the animal processed and cut (Do you want all hamburger? Or do you want steaks and roasts too and how thick do you want them cut? They'll do all the processing and call you when it's ready to pick up. It's usually frozen at this point, so you'll have to be sure to have a pretty big freezer to put it all in when you get home.
I can guarantee if you buy beef from a local farmer, you'll *never* want to eat store-bought ever.again!!!
Let me know if you have other questions -- as you can tell, I"m long winded and very willing to help! Us farmers need all the support we can get, especially from local customers who can actually draw that line between farm preservation support and good, local food.