I know I can divide the integers into 2 different infinite sets such that they form a partition of the integers, like the odd numbers, and the even numbers.
Is there another (trivial) way to partition the integers into $3,4$ or more different infinite sets?
If you want n partitions, just partition them by divisibly by n, so i-partition includes all the integers n_i such that n_i mod n = i.
Another way to divide in 2 partitions is to look at all the numbers that are greater or less than equal to say 0