When determining whether a ket-vector is normalized, should the ket be complex conjugated?

The condition for normalization for a ket vector is $$\langle A \mid A\rangle = 1.$$ However, to test if ket $$\mid A \rangle$$ is normalized, should I form the inner product with its complex conjugate $$\langle A^* \mid$$? How do I obtain the $$\langle A \mid$$ in the equation?

Here $$A$$ is just a quantum number (or a set of quantum numbers) labeling a vector, so it need not be conjugated. Products such as $$\langle A^*|A\rangle$$ do happen, e.g., when dealing with coherent states, but in this case $$|A\rangle$$ and $$|A^*\rangle$$ are different states.
The exact way of evaluating the product $$\langle A|A\rangle$$ depends on the representation that you work with. E.g., if these are two states in the coordinate representation, $$\varphi_n(x), \varphi_m(x)$$, we have $$\langle n|m\rangle = \int dx \varphi_n^*(x)\varphi_m(x).$$ If these are column vectors, then $$|\uparrow\rangle =\begin{bmatrix}\alpha\\\beta\end{bmatrix}, \langle\uparrow|=\begin{bmatrix}\alpha^*&\beta^*\end{bmatrix},$$ and $$\langle\uparrow|\uparrow\rangle = \begin{bmatrix}\alpha^*&\beta^*\end{bmatrix}\cdot \begin{bmatrix}\alpha\\ \beta\end{bmatrix}.$$
Finally, as it may have been clear by now from the examples, the normalization factor can be included as a simple numerical factor. I.e., if $$|A\rangle$$ is an unnormalized state, then its normalized version is $$|B\rangle=c|A\rangle$$, and this factor is indeed conjugated: $$\langle B|B\rangle = |c|^2\langle A|A\rangle=1 \Rightarrow |c|^2=\frac{1}{\langle A|A\rangle}.$$
The easy way to manipulate kets is to use the resolution of unity as much as possible $$\int d^3x |x\rangle \langle x| = 1$$ Similarly in momentum space $$\int d^3p |p\rangle \langle p| = 1$$ Then you have $$\langle A |A \rangle = \int d^3x \langle A|x\rangle \langle x|A \rangle$$
and since $$\langle A|x\rangle$$ is complex conjugate of $$\langle x|A \rangle$$ it is clear that complex conjugation applies to the wave functions, not to the quantum numbers