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 > As an exercise, I am planning to implement a Lisp interpreter.

I assume you mean to implement one in C or C++, in which case, don't
do that.  You'll end up with an understanding of how a Lisp
interpreter works & how to implement one in C or C++, but you won't
end up with an understanding of programming in Lisp.

To understand Lisp, you should program in it.  As you repeatedly
become amazed at how easy & simple it is to do things that'd be hard &
complex in C or C++ then you'll start understanding why Lisp has its
following.  When you start doing meta-programming in Lisp then
you'll start to really understand it.

Most lisp books are pretty decent, and others have recommended some.
But, I'd recommend Structure and Interpretation of Computer Programs
by Abelson and Sussman.  It's scheme & not common lisp, but it does a
good job of illustrating a variety of lisp programming paradigms.