I've actually gone looking for this exact thing a few times. Most Merlin fics kind of skip over the nitty-gritty of the magic itself and laser-focus on the reveal to Arthur or the romance, but there are some that really dig into the mechanics.
A great one I bookmarked ages ago is 'A Sort of Steady Falling' by an author named ivyblossom. It's on AO3. It's a modern AU where Merlin is a college student and Arthur is his TA, but the magic training aspect is so cleverly woven in. Merlin discovers old books in the library and has to figure things out through trial and error, with Gaius as a professor guiding him. The process feels very authentic—the frustration of spells fizzling, the thrill of a small success.
There's also a longer series, 'The Once and Future Queen' by BeautifulFiction, which isn't solely about training but has incredible sections where Merlin consciously works to understand his power, often using ancient texts Morgana finds. It treats magic like a muscle he needs to learn to flex properly, which I always prefer over him just being instinctively overpowered.
If you're okay with crossovers, 'Harry Potter and the Servant of Emrys' explores Merlin being thrown into the HP world and having to reconcile his innate power with structured spellwork. The contrast in magical systems is the whole point, and it's a fun read for that reason alone.