There are only three Medical Entrance Exam NEET, AIIMS & JIPMER. The admission into private colleges will be through MCI counselling on the basis of NEET score only. There is no any state level or university level entrance Exam  Or direct admission for Medical programme.

Good Luck!

Other than neet there is AIIMS (all India institute of medical sciences) which conducts their own test for admissions to mbbs every year and their cut off marks also goes very high. 

St John's Medical college also conducts separately their own exam for admissions.