The procedure, following parliaments based on the Westminster Model, is that the Governor, having taken advice, will invite the largest Parliamentary group (here the BJP) to attempt to form an adminstration.
As the BJP doesn't have a majority it would either have to enter an arrangement with another party, or depend on support from sufficient numbers of rebels from other parties. If after a certain amount of time (two weeks is often given) the largest party has been unable to form an administation, the governor (again acting on advice) would either invite the next largest group (INC in this case) to attempt to form an adminstration, or to call fresh elections.
I haven't followed this very carefully, but my understanding is that the BJP is unlikely to be able to achieve a majority, but a coalition of the INC and JDS could be formed. The details of the coalition would be a matter for the parties to decide. It could be a formal coalition, headed by the INC (as the major party of the coaltion) but with some ministeries given to the JDS, or a looser arrangement in which the JDS supports the INC in critical votes, but reserves the right to vote independently on other matters.
These matters can take several weeks to sort out, so a new adminstration is not likely to form quickly. The civil service remains in place and can ensure that the basic function of government is not disrupted too much.
On 17th May, the BJP were invited to attempt to form and administration. B. S. Yeddyurappa became Chief executive, but was unable to demonstrate a majority in the Assembly, he was removed after 6 days, and a coalition headed by H. D. Kumaraswamy of the JD(S) took over on 23rd of May.