You can apply here:
http://www.socialsecurity.gov/medicareonly/Just to clarify your EARNED benefit (don't ever let anyone convince you the government is "giving" you anything) begins on the first day of the month of your birthday. So for example if your birthday is March 18th your benefit begins 01 MARCH.
I highly recommend everyone apply at 64 years and 8 months (the earliest allowed). Sometimes (rare but it happens) there are record problems, incorrect data, enrollment issues. By applying early you will know in advanced that you are approved and the date your benefit begins. Then you simply need to wait until your benefit begins with confidence that you are covered.
About Part B:
Medicare Part B costs about $115 per month. If you aren't collecting SS yet Medicare will send you a bill. You can opt out of Part B and opt in at a later date but you *may* pay a penalty of roughly 10% per year you opt out. For example if you opt out of Part B for 5 years when you do Opt in your premium will be roughly 50% higher. There are exceptions depending on your situation so if you think you will want/need to opt out of part B be sure to contact Medicare.
About Medigap:
There are private plans which cover the "gsps" in medicare. I wouldn't recommend them but some like them. If you think you will want/need medigap coverage start research now. If you apply within 6 months of starting medicare then they can't disqualify you. They have to take all applicants (and at the same rate) that enroll within 6 months.