Are these isomorphic $\mathbb{Z}_{2}\times\mathbb{Z}_{3}$ and $\mathbb{Z}_{9}^{*}$ Is  $\mathbb{Z}_{2}\times\mathbb{Z}_{3}$ isomorphic to $\mathbb{Z}_{9}^{*}$
both have orders 6
both have elements with orders 1,2,3,6 (1 element of order 1, 2 elements of order 3, 1 element of order 2 and 2 elements of order 6) 
Both are cyclic thus Abelian
Can I assume then that they are isomorphic? Or a specific isomorphism must be constructed?

You've already got 99% of what you need to make an isomorphism anyhow, so you should :)
You noted that they both have elements of order $6$, so that they are both cyclic groups of order 6. Just send one generator to a generator of the other one and convince yourself it's an isomorphism.
