In almost all instance, questing is always a better source of experience than grinding. That said, there are times when it's better to kill every enemy between you and your quest goal instead of running around and ignoring them.
Be aware that you will need to either grind a bit or travel a great deal on top of doing quests while leveling. You won't be able to just sweep through quests hubs doing all the quests present and keep up. Know which other zones have quests close to your level so that you can swap between them when you feel like you're hitting a wall.
If you want to level quickly, always be killing things as you move. You can multitask this way.
You won't be able to simply pick up any weapon type and start swinging like a pro. You'll need to train in new weapon types at Weapon Masters in the main cities, and then level up your weapon skill first. Don't try to go into battle against an enemy near your level with a weapon type you've never used - raise your skill up first on lower level enemies.
If you swap to a new weapon type and need to level the weapon skill, you can spam instant low cost moves (e.g. hamstring) to count as a "swing" and have a chance to level. Also consider tipping a mage to give you Arcane Intellect, because intelligence increases your weapon skill level rate.
Although most classes can kill orange or even red mobs, it's usually neither easy nor fast. Most classes are most comfortable killing yellow mobs and looking to move on soon if you're killing green mobs.
Spirit is generally an extremely good stat while leveling, due to the downtime reduction.
WoW Classic Combat Tips (PvE)
You can't automatically queue up for dungeons or use finders to locate groups in Classic WoW. If there's a dungeon you're keen to explore, hang out in the trade cities or in the zone where the dungeon is to find people who are also looking for a group
Always have an escape route in mind when entering dangerous areas or strongholds full of monsters. It's easy to find yourself cut off by respawning enemies after carving a path inside. Pay attention to your surrounding so you don't accidentally invite disaster.
Know your classes strengths and weaknesses, and let that dictate the flow of battle. Hunters and Druids can take on much tougher enemies solo than a Rogue, so try to not to get too risky or you'll spend more time running back to your corpse than you'd like.
Depending on your class, keep an eye on how you're finishing off monsters before preparing for the next pull. There's no need to waste your combo points/rage/mana for a big attack on an enemy that's almost dead anyway - when it can be better spent preparing for the next enemy in line.
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