What paperwork do I need as support for a UK Standard Visitor visa application?

You are a Bangladeshi resident of Nigeria and you want to sit the PLAB2 exam in the UK. You propose to use a co-sponsor because (presumably) your own finances are weak.

My father, who is also a practicing doctor in Nigeria, will be my financial sponsor.

We have a great history of PLAB2 refusals in the archives here where well-to-do parents attempted to act as co-sponsors. This is the refusal formulae that the ECO frequently uses...

...I acknowledge your sponsor proposes to pay for your visit, however, it is the circumstances of the applicant that remain paramount when assessing your application...

What they are saying is that even when the parents offer to provide sponsorship, the application can fail because of the applicant's lifestyle and personal circumstances.

Will I need to submit my own bank statements alongside my father's bank statements while applying for the visa?

It is not necessary to submit your own bank statements if you have other evidence that can establish your lifestyle and personal circumstances as stable and sufficiently tied to the Nigerian economy. The problem is that there's often a shortfall in evidence in PLAB2 cases because the applicant is just starting out in life and has not yet achieved the social and economic standing that the British consulate requires. I suggest reading this article to see some of the things they will be looking for: Should I submit bank statements when applying for a UK Visa? What do they say about me?

Also, do I need to submit police verification report and TB clearance report for UK Standard Visitor visa?

As a Bangladeshi resident of Nigeria, you will need to have a tuberculosis test at either the Lagos or Abuja clinics. It will cost USD 145 over and above the application fee.

As for the police verification report, they do not require (or expect) this evidence. Sending one can be harmful because it proves you have not examined the Supporting Documents Guide and it's just another thing they have to send to the forgery detection unit. If they decide your application is complex they will contact you and tell you what else is needed.

Use the guidance to inform your application and check the archives to see where PLAB2 applications have been refused previously.