Unfortunately I only have one skill even though I am level 3 for some reason, and how am I supposed to increase my proficiency in alchemy and gardening?
Mdc94 has giving you the method how to learn new spells. Hint: if the candle in your study burns, there is something to learn.
Some spells have requirements. Krowly will mention them, but mostly only once. This game requires you to pay close attention to the dialogues.
Craft potions in your lab. When done several times, this will raise your alchemy proficiency.
Same goes for Cooking. Hint: help Alice in her home on mornings when it's raining. This will raise your relationship level and cooking skill (up to intermediate) at the same time. Later, help out Samantha. This gives a lot of XP and is repeatable.
Gardening, work in your garden (use the shovel that's lying on your field) on non-rainy mornings or afternoons.
Finally, did you know there is a comprehensive walkthrough posted in the OP ?