Critical Hits
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.
Spells
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
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.
Advantage/Disadvantage
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.
d100
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.
Class Features
Bardic Inspiration
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.
Superiority Dice
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).
Divine Smite
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.
Colossus Slayer
Hunter Rangers can take this option to deal an additional d8 of damage to already-injured targets.
Sneak Attack
Rogues will often deal additional damage equal to half their level in d6s (rounded up).
Wild Magic
Wild Magic Sorcerers will probably roll a d100 a few times each session to determine the additional effects of their spellcasting.
(5 × level) + INT mod
Because no parentheses or other mathematical indicators are given in the Hit Points section of the Homunculus stat block,
Hit Points equal to five times your level in this class + your Intelligence modifier
directly translates to "HP = 5 × [your level in this class] + [your Intelligence modifier]", which in turn means "HP = (5 × [your level in this class]) + [your Intelligence modifier]", because of the order of operations.
Rubiksmoose also notes that Jeremy Crawford has given guidance on similar subjects. One of his tweets is here, addressing the Arcane Ward feature for Abjuration Wizards:
[...] The ward has hit points equal to twice your wizard level + your Intelligence modifier.
Best Answer
I've played exactly one session with my 3rd level artificer (edit: this was with the February 2019 edition of the artillerist, not the newer version released in May) so your mileage may vary but here's what I have found so far. The session I played is exactly what you described, 3 encounters with a climactic end battle.
The thing with the artificer is that they're flexible but also limited in what you can do in a given day, especially at low levels. The turret is only free for one casting of it, after which you’re using spell slots. At 3rd level you only have 3 slots, so it really feels quite limiting if you’re planning on casting any other spells at all. For my session I barely squeaked by after using Shield, a second turret, and one other spell. The second turret was key in the final battle and kind of saved our collective bacon because we barely scraped by. At one point I was the only party member left standing after the rogue and sorcerer went down and our paladin had his intelligence drained by an intellect devourer. But the force turret was able to help me deal extra damage when I needed it and push the enemies away from our sorcerer when I needed to as well.
I wouldn't be worried about the magic items. Especially at low levels the number of magic items you could possibly create are limited to two at a time. I used the enhanced armor to boost my AC, and a bag of holding because they're always good, but a lot of the other stuff in the replicate magic items table is very situational. In hindsight for my encounter I could have used the cloak of the manta ray, but I didn't see that coming and didn't take that infusion, so I had to get by without it.
All told I think the artificer is a great class but I don't feel like I was overpowered compared to anyone else in our party. I really felt like I had to be judicious in the use of my powers because you don't have enough to be using them all the time.