Constructible enumerable set
Hints:


For $S_1 \cup S_2$, use the same idea in the proof that the natural numbers and the integers have the same cardinality. The proof goes by listing the integers in sequence:
$$ 0,1,-1,2,-2,3,-3,\ldots $$
You can think of $S_1$ as $0,1,2,3,\ldots$ and of $S_2$ as $-1,-2,-3,\ldots$.
For $S_1 \times S_2$, use the same idea in the proof that the natural numbers and the positive rationals have the same cardinality. The proof uses the "diagonal method", and consists of listing the rationals sorted by the maximum of the numerator and the denominator:
$$ \frac{1}{1}, \frac{1}{2}, \frac{2}{1}, \frac{2}{2}, \frac{1}{3}, \frac{2}{3}, \frac{3}{1}, \frac{3}{2}, \frac{3}{3}, \ldots $$
In this case $S_1=S_2$ are both $1,2,3,\ldots$.