I am using a tool which backsup data from my SFDC instance to local MySQL database. I am concerned with Account and Opportunity object only. But I want to bring all those other objects which are related in any way to Account and Opportunity. Is there any way to determine what objects do Account or Opportunity are related to via any custom or standard field?
You can use a variety of free metadata search tools like Octopus to see what objects and fields might be related. You can also directly do this by using Schema Explorer in Setup to see what objects are related to Account and Opportunity via lookup relationships. The issue there is that some my have multiple paths that continue beyond the first look-up; particularly with opportunity line items, products and quotes which are things that may not be inherently obvious. I suggest you also take a look at this ERD for Sales Objects and some of the ones that follow in the same document.