Because you did not also sign up for Part B when you turned 65 and were not working, you are eligible only during January thru March (General Enrollment Period/GEP) to request Part B by contacting Social Security. Social Security will delay your effective date for Part B until July 1 of that year. You may also be required to pay a late enrollment penalty of 10% on top of the current Part B premium for each 12-month period you could have had Part B but did not take it.
Once you have both Part A and Part B, you can elect to go onto a Medicare Advantage/HMO plan or a Medicare Supplement/Medigap policy and stand-along drug plan to be effective 7/1.
Also, if, when you became eligible for Part A, you did not enroll in a drug plan and do not have creditable coverage for drugs, you will incur a penalty of 1% of the average national premium for every month you were eligible and did not enroll. This amount is added onto your drug plan premium in perpetuity.
If, however, you have been working since turning 65, you can apply for Medicare Part B with no penalty. There are 2 forms that need to be completed (one by you and one by your employer; the employer form needs to indicate a termination date of your employer health plan coverage); both forms need to be submitted to Social Security to activate your Part B. If your coverage under the employer plan will terminate on April 3oth, then you are requesting Part B coverage as of May 1st.