First up, welcome to RuneScape! It's great to have you here, and hope you're enjoying yourself.
Quests are my favourite part of the game, and there's a whole smorgasbord of them to choose from (as you point out), from funny, irreverent short quests to epic narratives. The quest pathing system will give you a good grounding in what to do at which point, but here are some of my favourite early quests - a mix of serious, one-offs, first of quest lines, and fun ones:
- Missing, Presumed Death (kick off to Sliske quest line)
- Song from the Depths (we actually did a live play-through of this recently - great audio)
- Broken Home (fantastic one-off horror-style quest)
- Vampyre Slayer (classic)
- Priest In Peril (entry to Morytania and Vampyre quest line)
- Plague City (kicks off the Elven quest line, which eventually leads to unlocking Prifddinas)
- A Soul's Bane (bit of a dated quest now, but interesting concept)
- One Piercing Note (fantastic audio, murder mystery-style)
Hope this helps!