Keep in mind that these quests are optional, and are not necessary to continue leveling and capturing companions. They are also account-wide, so if you complete the quest chain on one character, you will not be able to complete it again on another. However, any progress that you've made will be saved and you can pick up where you left off in the quest chain on another character. The dailies that you've unlocked will also be accessible by all your toons.
Mounts also take up space in your bags, bags which will of course be much smaller. Every bag space counts. With no guild banks, and much smaller storage space, your bank alts are going to be essential. You’ll need to remember to always post everything back to a bank alt, and to remember to get everything from the mail box to the Auction House. Which is only in Ironforge or Orgrimmar.
After choosing a race, class and customizing the look of your character, you will spawn in your race's starting zone, usually within a small settlement. Typically, these beginning areas are designed to get you to around level 10 before you leave and move on to the next zone. Right away, you will notice that some NPCs have a yellow exclamation mark () over their head. These symbols will also display on the mini map in the top-right corner of the screen if you are close to them. Talking with these NPCs will allow you to read and accept a quest they have for you. The objectives and status of this quest can be checked at any time by opening the quest log (L). Once you have completed the objectives for said quest, you can return to the NPC (or possibly a different one) to turn in the quest. The NPC whom you need to turn a quest into will have a yellow question mark () over their head. As you complete quests and your character gains levels, always be on the lookout for new quests being offered by NPCs.
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.