All things considerered you should stick to the Sorcerer
Choosing a spellcaster
There are 8 classes with Spellcasting ability.
Additionaly, there are 2 subclasses.
Excluding non-Arcane classes and Eldritch Knight (based on information in your question) leaves us with the following list:
- Bard
- Rogue, Arcane Trickster
- Sorcerer
- Warlock[1]
- Wizard
Spells of 2nd level and lower listed below under each of the class' section are combat-oriented and require neither spell attack nor saving throw from a target nor use spellcasting modifier for additional effect, such as healing spells. Recommended spells will be emphasised.
Bard
Personaly, I realy like the idea of a Fighter/Bard multiclass, but class' main feature, Bardic Inspiration, heavily relies on Charisma: the number of Inspiration die depends on your Charisma modifier, on top of this some of the Bard's subclasses' features rely on a saving throw from your target. I personaly recommend College of Swords for your character. Multiclassing into Bard will net you additonal proficiencies in one skill (any) and musical instrument. Bard's spells:
- Cantrips: Blade Ward, True Strike
- 1st level: Bane, Longstrider, Sleep
- 2nd level: Cloud of Daggers, Enhance Ability, Heat Metal, Invisibility, Silence, Warding Wind
Rogue, Arcane Trickster
Spell list is limited and spell slot progression is 1/3 of a regular caster. Extra 2d6 from Sneak Attack is handy but heavily relies on a weapon you use. Unless you're Rapier&Board fighter that will utilize Color Spray or Sleep during combat, I recommend to skip it. Multiclassing into Rogue grants you proficiencies in one skill (rogue's skill list) and thieves' tools. Rogue's spells:
- Cantrips: Blade Ward, True Strike
- 1st level: Color Spray, Sleep
Sorcerer
@DuckTapeAl already has a good answer. In addition, you can cast Booming Blade using Twinned Spell Metamagic ability to make 2 melee attacks instead of one untill you get Fighter's Extra Attack.
- Cantrips: Blade Ward, Booming Blade, Green-Flame Blade, True Strike
- 1st level: Color Spray, Expeditious Retreat, False Life, Magic Missile, Shield, Sleep
- 2nd level: Alter Self, Blur, Cloud of Daggers, Enhance Ability, Enlarge/Reduce, Invisibility, Levitate, Mirror Image, Misty Step, Shadow Blade, Warding Wind
- Cantrips (cleric): Guidance, Resistance
- 1st level (cleric): Bane, Bless, Divine Favor, Protection from Evil and Good, Shield of Faith
- 2nd level (cleric): Aid, Protection from Poison, Silence
Warlock
A solid pick, since both Superiority die and Warlock's spell slots restore on a short rest. Invocations are handy and Pact of the Blade gives you a magical weapon. Choice of the patron relies on your combat behaviour and expended spell list you might like but I personaly recommend Fiend for you.
- Cantrips: Blade Ward, Booming Blade, Green-Flame Blade, True Strike
- 1st level: Armor of Agathys, Expeditious Retreat, False Life, Hex, Protection from Evil and Good, Sleep
- 2nd level: Cloud of Daggers, Invisibility, Mirror Image, Misty Step, Shadow Blade
- Suitable Invocations: Fiendish Vigor, Thief of Five Fates, Improved Pact Weapon (Pact of Blade exclusive)
Wizard
Wizard's Arcane Recovery feature allows him to regain spell slots once per day on a short rest. There are numerous Wizard's subclasses but there are few that realy suits you, these are School of Divination (for two d20 you can use to replace nearly any roll in your line of sight) and War Magic (for +2 AC or +4 to a saving throw on a reaction).
- Cantrips: Blade Ward, Booming Blade, Green-Flame Blade, True Strike
- 1st level: Absorb Elements, Expeditious Retreat, False Life, Find Familiar[2], Longstrider, Magic Missile, Protection from Evil and Good, Shield, Sleep
- 2nd level: Alter Self, Blur, Invisibility, Levitate, Magic Weapon, Mirror Image, Misty Step, Shadow Blade
[1] - Pact Magic isn't a Spellcasting per se, but I count it as one in the scope of this question.
[2] - Using an owl familiar might grant you an advantage nearly on every turn. Also, Find Familiar is a ritual spell so you do not spend any resources.