For the duration of the fight, creatures in the immediate area will not be visible to you (but they may still attack you if they wander too close). Keep in mind that other players will still be able to see you and attack you if you are eligible for player vs. player (PVP). Other players will also be able to view your pet battle in real time, including the pets involved as well as their health as the match progresses.
I am currently working full time on my Vanilla WoW leveling guides. Soon there will be a 1-60 speed leveling guide for Alliance, along with in-game guide versions available for my members area. As of right now, there is no members area, but if you would like to show support for my efforts into this project, you can simply donate to me via Patreon if you like (you will get some cool rewards in return). Or you can donate directly with PayPal if you like. All donations are highly appreciated and they will only motivate me more to continue making the best vanilla WoW leveling guides on the web.
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 is a great advantage to start making some silver and your first gold coin at low levels already. Skinning is a nice choice here. Fast and easy way to gather resources, leather of all kinds while leveling for those who are at the race to level 60. There are many good spots to grind beast mobs and skin them. I like doing it as a Troll Hunter, trolls have a passive racial ability: + 5% bonus damage against beasts.
This is another one that will most likely be a boon rather than a bane to the player base, though time will tell just how 14years of experience will affect the economy of Classic WoW. There has been no concrete word on just how AddOns will work in Classic, but if the infrastructure of the game works the same as it does in retail, there is a good chance that most of the mods that work in retail will work in Classic. This means that quite a few people will be running around with a whole host of gathering, crafting, auctioning, and gold making mods. Now, those mods did exist back in Vanilla, but not in the same way they do now, and not as many people had them back then as will in Classic. This will drastically affect how effective the auction house will be, and hopefully will affect the economy as a whole in a positive way. Another thing that will most likely see a large increase in popularity is carry runs. These have steadily grown in popularity since Vanilla, and rest assured with the old 40 man raid size that there will be quite a few “Molten Core full carry master loot ON PST for prices GOLD ONLY” being spammed in trade chat. Whether tokens will be available in Classic has yet to be discussed, but if so it will have an enormous impact on the economy of Classic. This, in addition to the differences I will cover in the next section, will have a pretty large impact on the endgame of WoW Classic.
The simple truth behind all of this bickering and squabbling is that there really isn't a bad way to enjoy a game. You like the feeling of Classic taking you back to your young(er) days and that rush of falling in love with WoW for the first time? You like the slower pace of both combat and leveling? You like that the social aspects of the game take a front seat? Good, enjoy it! You prefer modern WoW and BfA with it's tons of improvements, smoother design and more player-friendly features? Great, enjoy it! It doesn't matter whether you prefer modern WoW or Classic or WotLK or whichever your personal high-point was (it was Burning Crusade and Legion for me), you're not wrong. You literally can't be. No matter what anyone says in all these heated discussions, they care about WoW, in whatever form they prefer it. So why can't this be enough of a commonality for us to have a civil discourse?