[RPG] Does a cantrip need to be learned like level 1-9 spells as a wizard

Change how you play your wizard so those are the right spells

With the spells you've listed you can play an effective wizard but it will likely require changing up your play style. Let's look at some tactics that suit your spells.

Blast 'em

This isn't going to be your strong suit but there are plenty of opportunities to contribute with some straight up blasting. You have a variety of damage spells that target different enemy weaknesses.

• Lumbering, low dex save enemy?
  • Acid splash, flaming sphere
• Heavily armored tank?
  • Magic missile
• Wispy, fey thing?
  • Poison Spray
• Average joe?
  • Fire bolt

Remember that you can cast the Magic Missile and Flaming Sphere as higher level spells to maximize their damage when it counts.

Lockdown

For the rest of the party to do damage they need to stay alive. The best way to keep them alive is to keep the enemy from being able to attack them. You excel in this area.

• A bunch of archers with cover

  • Fog Cloud is incredible here. It's 120ft range means you may be able to cast it without getting into the archer's effective range, and once it's on top of them the rest of your team can close safely. Dropping it on top of the enemy cover can also force them to expose themselves to the ranged members of your team. Alternatively you can use Fog Cloud as temporary cover for your team allowing thieves to hide, other casters to buff, or just forcing the enemy to close with you.

• Party is out numbered or flanked

  • Grease/Web/Stinking Cloud are so good. You pick some part of the battle and say, "That part sucks now". Web/Stinking Cloud can lock groups enemies out of the fight functionally turning one Deadly encounter into two Easy ones. Pick the best spell depending on what saves you think the enemies will fail.

• Enemy is really hard to hit

  • We already talked about Web but it's worth talking about again. Enemies caught in a Web end up restrained which gives your allies advantage on their attacks. Against high AC enemies this can be crucial. You can also follow up with your Fire Bolt for an interesting combo.

Your remaining spells have solid general utility as well.

• Enemy is hard to get to? Make the Fighter fly.
• Enemy rushing you to try and break your concentration? Misty step to safety, and since it's a bonus action follow up with something.
• You captured a Kobold but he doesn't want to tell you about all the traps upahead? Suggest he talks to save his skin.

Overall you have a wide range of spells and should be able to find a use for them in most situations.

Normally, you could not cast it at all with no free hands- the spell has V, S components, which means you need an empty hand to perform the somatic components, even if one of the hands is holding a focus.

However, the War Caster feat exists. So what happens when you cast it with full hands in that scenario?

Well, the spell says the flame appears in your hand and doesn't hurt you or your equipment. It also says it remains there for the duration, and lists the ways the spell can end. Since the spell effectively states the flame remains in your hand until the spell ends or the flame is hurled, I think the magical flame is in no danger of being doused, smothered, or dropped just because something else is in your hand- the spell doesn't allow for that in its text.

You should be safe to continue holding your equipment in-hand, and the flame will continue to co-exist/(safely) burn around it.

As far as an opportunity attack via War Caster goes, War Caster only adds the ability to cast a spell as a reaction as an alternative to the usual opportunity attack. RAW, I believe it would allow you to newly cast produce flame and then immediately attack with it ("When you cast this spell, [...] you can hurl the flame at a creature within 30 feet of you"), but wouldn't allow throwing a previously-prepared instance already in your hand- not that it matters, you could just cast it anew instead for no real difference.

Do note that using the War Caster reaction to attack with the spell would result in attacking with disadvantage, as the target would still be within 5 feet of you when you made the ranged spell attack.