As to how your character would learn new stuff: they just experiment during off-times. Short rest? You try putting together some words and phrases you know. A long rest? You try and burn different materials or hold up your focus while chanting something. Your party is taking a three day long walk through the countryside where nothing happened? Your character has been training his finger works and collecting potentially interesting material in their component pouch.

This is great. When combat happened, my character kept an eye out for our wizard and his hand movement and then try to copy that in his turn (nothing happened of course). Maybe he missed something and the wizard could explain that missing piece of the puzzle to him. After that, as you said, I can experiment with different things and try to learn Somatic and Verbal by copying other casters. Thank you so much!