There are two parts of the quest chain for each continent. The first part involves defeating numerous Master Pet Tamers, and the second requires you to triumph over the Grand Master Tamer. After you've bested the Grand Master Tamer, dailies from each Tamer will unlock for that continent. All the dailies continue to reward experience for your companions, however only the Grand Master daily quest will reward the Sack of Pet Supplies.
Zygor's Leveling Guide for The Elder Scrolls Online (PC Version) is an in-game, software based, strategy guide that will show you the quickest way to level your characters from 1-50. You can view and use the guide directly inside the game without having to print out anything or view the guide in a browser. This also means that our guides are dynamic and can track your progress in realtime.
Only really to expand on Gráinne's comment as regards the 'collecting' and 'roster development' sections - which are, as said 'distinct' although they are intrinsically linked - I would like to point you in the direction of the in-game Achievements > Battle Pets > Collect tab. There are a series of achievements called Raiding With Leashes, pretty much all of which can be completed via the auction house - if you have the gold/inclination - or by completing old raids.
It was the experience of the game that brought pleasure and joy, and that's what a lot of us want again. Not the numbers. When I play Classic I want to go through the deadmines with people I like and overcome a challenge together and see myself become stronger as a result. I don't want to play Classic so I can say "booyah! That number is exactly the same as it was 13 years ago!"