I was wanting to build a character that was raised by a warlock before going out on their adventuring days. My question is if the character wants to cast spells like the caretaker warlock, which would be a closer fit?
Either can work thematically.
Take this description from the Warlock class:
And sometimes, while poring over tomes of forbidden lore, a brilliant but crazed student’s mind is opened to realities beyond the material world and to the alien beings that dwell in the outer void.
Poring over tomes? A student of magic? It certainly sounds like a novice Wizard's forbidden studies, compare to this description from the Wizard class:
They can also learn them from other wizards, from ancient tomes or inscriptions, and from ancient creatures (such as the fey) that are steeped in magic.
I could imagine a Warlock teacher showing a Wizard student some forbidden tomes and studying together.
Drive is also a common thread between the Warlock and the Sorcerer:
No one makes a pact with such a mighty patron if he or she doesn’t intend to use the power thus gained. Rather, the vast majority of warlocks spend their days in active pursuit of their goals, which typically means some kind of adventuring.
People with magical power seething in their veins soon discover that the power doesn’t like to stay quiet. A sorcerer’s magic wants to be wielded, and it has a tendency to spill out in unpredictable ways if it isn’t called on.
Sorcerers often have obscure or quixotic motivations driving them to adventure. Some seek a greater understanding of the magical force that infuses them, or the answer to the mystery of its origin. Others hope to find a way to get rid of it, or to unleash its full potential.
A Warlock teacher may train a Sorcerer student on the nature of their innate power. There could be a lot of life learning about striving towards your goals that a Warlock could impart onto a Sorcerer student.
Font of Magic can emulate Warlock casting
I think that "in game" Sorcerers more closely match Warlocks. The defining feature of Warlock spell casting is few spells, but powerful spells. A Sorcerer can Font of Magic to transmute their lower level slots to higher level slots.
For example a Warlock 5 has two 3rd level slots (effectively six 3rd level slots for the day). A Sorcerer 5 has two 3rd, three 2nd, and four 1st. By cannibalizing their lower level slots they can generate a total of 17 SP, which can generate three more third level slots - giving them effectively five third level slots (and one first level) for the day. This is very similar to a Warlock's six slots.
A Wizard would not be able to emulate this, so I think that a Sorcerer is a closer match. Someone who was born as a Sorcerer but trained by a Warlock would probably think in this way.