I think the best website is Ancestry.com. Some of their links are free to access all the links you have to subscribe.
http://www.civilwardata.com/ is solely Civil War research but nothing is free. You actually sign up for this one tell them I sent you. I get a reward for referrals.
http://www.itd.nps.gov/cwss/ is free but you only get bare bones information on soldiers. Its the quickest way get a list of soldiers & their units as a staring point.
Generally, companies did not operate independently of regiments so whatever a soldier wrote of his experiences applied to the regiment.
There is an Ohio in the Civil War website with brief regimental histories that will give you an outline history of units you are interested in.
Also feel free to contact me with a specific research request.
I'll see what I can do.