How do people upgrade/grow their inventories?

I am seriously at a dead end of trying to figure out how on earth people can get these high tier and sought after pets within such a short span of time, without cheating and using less than legal sites.

I desperately need advice for upgrading/growing my inventory. What are the basics? Is there anything I should avoid or look for? Where do I even begin? Any insights would be greatly appreciated fron me.