Yes , hes taken out actual trained assassins and martial artists with his base skills and spider sense. It was always just chaotic. Cap said he faught like a spastic monkey when he was younger.
Now the fact he actually has training in his own style only makes him better.
During Spider-Island, Spider-Man went to Shang-Chi for martial arts training on the recommendation from the new Madame Web to compensate for the loss of his Spider-Sense. Together they created the Way of the Spider, a modified version of Peter's own combat style and Shang-Chi's martial arts.
So he does have some formal training, but mainly still relies upon his physical prowess?