After an initial signature drive, a proposed statute goes to the legislature for consideration. If the legislature kills it, or does nothing, or amends it in a way not acceptable to the petitioners, a second signature drive can put it on the ballot. Immediately after a law is passed, petitioners can put it on the ballot with a signature drive. To think through these choices, read first the primary applicable provisions of the Ohio Constitution, especially Article II, Sec. Reading these sections is no substitute for competent legal advice, which you must have prior to deciding to begin a campaign.
Election laws are complicated, changing and unforgiving.
It depends on what kind of ballot issue you are going for. After you turn in the signatures, the boards of elections will scrutinize them for errors. Some signatures and some whole petitions will be rejected. The legal requirements are 'net' signatures, that is, the number of signature declared valid by the boards of elections after they have checked the petitions. How many gross signatures you need depends on how you gather them.