There was more incentive to be out in the world in Classic mainly because a lot of people were still leveling their first characters and alts. I started playing in April 2005 and didn't get my first character to max level until somewhere around Summer 2006. It took quite a while longer to level then, and required going through most of the content for any given level range, which may have involved group quests and elite areas (Jintha'Alor, anyone?).
Depending on what point in time you're talking about, players had more or less incentive to go out into the world at max level. There was a time when the LFG channel was global, but otherwise finding groups required players to hang out in cities and spam LFG chat, or try to use the Meeting Stones, which rarely worked and were mostly reserved for last-minute attempts to salvage dungeon runs after a player left to eat dinner or raid with their guild.
For PVP, for quite a while you had to physically queue in front of the instance portals at the BG entrances. After they added Battlemasters you'd go to a capital and queue in front of them. There were also the PVP objectives in EPL and Silithus, though I don't think anyone ever bothered with the one in Silithus.
Otherwise, you'd go out into the world for the usual stuff: farming materials, including rare crafting materials that only dropped from certain mobs (devilsaur hide, felcloth, etc.). Farming mobs for rep or turn-ins (Timbermaw, Argent Dawn, etc.). Working on the extremely long endgame quest chains, like the dungeon set upgrade quests. There were also a lot of other weird little things you could do, like collecting items for the various summonable bosses in Silithus.
Honestly, looking back, I can barely imagine how I managed to spend as much time playing WoW as I did. I suppose I mainly did PVP back in Classic and BC, which was generally more compelling to me than "kill thousands and thousands of furbolgs so you can get that trinket that summons a furbolg to help you fight for a few minutes."