I came in at the tail end of Classic and remember a lot of things about the first time I rolled a character on Azeroth. Here are some of my memories about "the good ole days". Feel free to add to them with your own:
1. Mounts - Walked til lvl 40. Ran from South Shore to SM everyday to grind up to 40 and have enough gold to pay for my nightsaber and the training to go along with it. 300g was a lot of Money back then.
2. Basic Campfire - Had to carry Simple Wood and Flint & Tinder just to make a fire to cook with.
3. Hunter Pets - Jumping down in Wailing Caverns or Blackrock Depths and having the pet go the long route and pull 200 mobs. Good times.
4. Gathering - Nothing more fun than getting 3 fails at mining, skinning, or herbing before finally getting 1 of anything. Would take a whole day just to gather a hundred ore.
5. Flight paths - There were none in Elwynn Forest and only 1 in each zone. There was no FP or Stormwind Charger to go from Goldshire to Eastvale logging.
6. Special items - Getting all the way to Zul Farrak and forgetting the Mallet to summon the boss.
7. Addons - Back when Thottbot, Questhelper, and Cartographer were the only thing that told you what you were looking for and where you might find it.
8. Bags - As a Hunter, I gave up a bag slot for a quiver and would occasionally run out of arrows and have to use my Axes til I got back to a vendor.
9. Leveling - No BoAs, no RaF, no SoR, and no Guild perks. The XP you got was either regular or rested. There was no bump in XP gain.
10. Weapon/Unarmed skills - Everytime you'd switch from one weapon type to the other, you would have to equip the weapon and go level your skill with it. Also, remembering to level your unarmed skill, so when you go disarmed in a fight, you could still at least do damage.
There are probably another hundred things I remember about starting out, but I just can't sit and list them all. Fun times