What spellcaster multiclass options exist for a level 3 tiefling paladin, oath of vengeance, ideally the character should be able to still wear armour and have access to more spells/spell-slots than a normal paladin.
Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.
Charisma based casters...
Your paladin likely already has a decently high charisma, as many of their attributes scale of charisma. As a result, you will want to multiclass into a charisma-scaling caster, which leaves you with three good options.
...Sorcerer, Bard or Warlock
All these classes scale with charisma. You can already wear armor as a paladin, you will not lose this ability if you multiclass into something else. There is however one thing you will need to remember:
Somatic & Material components
Many spells require material or somatic components, which makes wearing a shield difficult for your multiclassed paladin. The War Caster feat will bypass this problem for somatic components, but the material component will remain an issue, and you'll either want to not have two occupied hands, or pick only spells that lack a material component.
Sorcerer, Warlock or Bard
is kind of the default option, with very good synergies. Quicken Spell (Metamagic) is huge.
gives you even more smites, depending on how many short rests your DM allowes.
gives you the best spell options, including Healing Word.
You seem to be eager to take a different class, so multiclass at the next opportunity. However, after that I would return to Paladin until I get Aura of Protection. It is that good. Also Extra Attack would be sorely missed.