I would like to write a program that creates a relational database that handles queries using only C++. What would be the most appropriate method? Using arrays, linked lists or something else? Please advise.
All the above. If you are new to c++ then you are not ready to tackle that project -- companies have spent years of manhours on that sort of project and those programmers are highly experienced people. database programs do not bring the entire files into memory because the files can get very very huge. Instead they use key files which contain key fields, offsets into the actual data file, and pointers to the next key field. This normally involves complex binary trees and structures.
Writing a small relational database program might be a good project for a 4th year college student (400 or 500 level college course).