When you score a critical hit (roll a natural 20 on the attack roll), you roll double damage. Therefore, you'll probably want to bring a second weapon die, depending on what weapon you'll be using. Greatswords and mauls are a 2d6 weapon, so you may want four d6s if you plan to use one of those.
Half-orcs and barbarians each have a feature that grants one extra weapon die on a critical hit, so you may want another one extra if you'll play a half-orc or barbarian, or another two extra if you'll play a half-orc barbarian.
For clarity, a weapon's die can be a d4, d6, d8, d10, d12, or 2d6.
Damage-dealing spells like fireball usually require several dice to be rolled. Most spells require a number of dice equal to the spell level plus one or two. Fireball and lightning bolt are notable exceptions; they each are 3rd-level spells, but deal 8d6 damage.
Spells with attack rolls can score critical hits, but most spells that require attack rolls are cantrips. Spells that provide a save DC (like fireball) cannot score critical hits.
Spells tend to use d6s and d8s, but some (like magic missile) use d4s and some (like inflict wounds) use d10s.
Spell damage does not passively increase as your character gains levels, but lower level spells can be cast with higher level slots. This usually adds one damage die per increased slot level.
Cantrips do increase passively as your character gains levels (unrelated to which class, in case of multiclassing). Cantrips all deal one die of damage from levels 1-4, two dice from levels 5-10, three dice from levels 11-16, and four dice from levels 17-20. Most cantrips require attack rolls and therefore can score critical hits. Most cantrips deal 1d6 or 1d8 damage, but some (vicious mockery) deal 1d4 and some (firebolt and eldritch blast) deal 1d10.
Most situational modifiers have been replaced with the advantage/disadvantage mechanic. For advantage events you will roll two d20s and take the better, for disadvantage you will roll two d20s and take the worse. These are frequent enough that you will certainly want to bring two d20s.
Occasionally, you may need to roll two d10s to emulate a d100 roll. Unless you are the DM or are playing a Wild Magic sorcerer, this event will be extremely rare.
Bards can distribute up to 5 bardic inspiration dice to their allies. These dice are d6s, but they upgrade a die size (d8s, d10s, then finally d12s) every five levels.
Battlemaster Fighters have special abilities that add a die to something (usually an attack or damage roll). At first you need four d8s, but at the number and size of the dice increase as you level (capping out at six d12s).
Paladins can expend a spell slot to deal additional damage on a melee weapon attack. Like many spells, the damage equals a number of d8s equal to the spell level plus one. It deals an additional d8 of damage to undead and fiends.
Hunter Rangers can take this option to deal an additional d8 of damage to already-injured targets.
Rogues will often deal additional damage equal to half their level in d6s (rounded up).
Wild Magic Sorcerers will probably roll a d100 a few times each session to determine the additional effects of their spellcasting.