What if you nested the MemberExpressionVisitor class inside another class? Since I like extension methods, create a static class that contains an extension method on Expression<Func<T,TRes>> that creates the visitor, runs it and returns the result:

static class LambdaExt {
    public static List<string> GetAccessedMemberNames<T, TRes>(this Expression<Func<T, TRes>> startExpr) {
        var v = new MemberExpressionVisitor<T, TRes>();
        v.Visit(startExpr);

        return v.GetAccessedMembers();
    }

    // put MemberExpressionVisitor definition here
}

Now you can use it like:

Console.WriteLine(String.Join(", ", exp.GetAccessedMemberNames()));

What if you nested the MemberExpressionVisitor class inside another class? Since I like extension methods, create a static class that contains an extension method on Expression<Func<T,TRes>> that creates the visitor, runs it and returns the result:

static class LambdaExt {
    public static List<string> GetAccessedMemberNames<T, TRes>(this Expression<Func<T, TRes>> startExpr) {
        var v = new MemberExpressionVisitor<T, TRes>();
        v.Visit(startExpr);

        return v.GetAccessedMembers();
    }

    // put MemberExpressionVisitor definition here
}

Now you can use it like:

Console.WriteLine(String.Join(", ", exp.GetAccessedMemberNames()));

Alternatively, you can add a constructor to MemberExpressionVisitor:

public MemberExpressionVisitor(Expression<Func<T1, T2>> startExpression) {
    originalExpression = startExpression;
    Visit(startExpression);
}

remove the Visit override entirely, and simplify the extension method to:

public static List<string> GetAccessedMemberNames<T, TRes>(this Expression<Func<T, TRes>> startExpr) =>
    new MemberExpressionVisitor<T,TRes>(startExpr).GetAccessedMembers();

This has the compiler enforce type safety, assuming you know your lambda expression variables types properly.