which items or achievements can get me pet slots?

There aren't any. You can buy pet slots in the Millionaire Center for scaling amounts of sP (starting at 20m and ending at 200m for the 10th slot) or for 2000 CSC each on your Pets page. You can choose a from your Loyalty Box on the anniversary of creating your account every year from 2 years on, and sometimes they're given out from site events (like the recent Quest-a-Thons) but those are tied to the unique events and you can't get them retroactively