A nice set of generators for the automorphism group of a finite abelian group is described by Garrett Birkhoff in his paper titled "Subgroups of abelian groups", Proc. London Math. Soc., s2-38(1):385-401, 1935.
Note that each finite abelian group is the product of its $p$-primary subgroups. An automorphism preserves the primary parts. So it is sufficient to consider $p$-abelian groups.
It is convenient to think of automorphisms of finite abelian groups as integer matrices. Expressing the group $A = \mathbf Z/p^{\lambda_1}\oplus \dotsb \oplus \mathbf Z/p^{\lambda^n}$ as a quotient of the free abelian group $\mathbf Z^n$, lift an automorphism $\phi$ of $A$ to an automorphism $\tilde\phi$ of $\mathbf Z^n$: $$ \begin{matrix} \mathbf Z^n & \xrightarrow{\tilde \phi} & \mathbf Z^n\\ \downarrow & &\downarrow\\ A & \xrightarrow{\phi} & A \end{matrix} $$ The matrix $(\phi_{ij})$ representing $\tilde\phi$ is an invertible integer matrix. As far as the automorphism $\phi$ is concerned, the value of its entries in the $i$th row are in $\mathbf Z/p^{\lambda_i}\mathbf Z$. Also $\phi_{ij}$ is $j$ dvisible by $p^{\max(0, \lambda_j-\lambda_i)}$. With this matrix representation, it is easy to do calculations. For example, composition is matrix multiplication.
As for yout query concerning cardinalities: if $\lambda_1>\lambda_2>\dotsc>\lambda_n$ and $$ A = (\mathbf Z/p^{\lambda_1}\mathbf Z)^{\oplus m_1}\oplus\dotsc \oplus (\mathbf Z/p^{\lambda_n}\mathbf Z)^{\oplus m_n}, $$ it is possible to deduce from the above description of the automorphism group that $$ |\mathrm{Aut}(A)| = q^{\sum_{i,j}m_im_j\min(\lambda_i,\lambda_j)}\prod_{k=1}^n \prod_{l=1}^{m_k} (1 - q^{-l}). $$ The group $\prod_{k=1}^n GL_{m_k}(\mathbf Z/p\mathbf Z)$ is a quotient of $\mathrm{Aut}(A)$ by a $p$-group.