My character is a 2nd-level Wizard without a familiar. I think it would be cool to get a familiar, but how does one turn an everyday animal companion into a familiar?

It's a specific ritual which varies according to setting. It's not written anywhere, but as a DM, I would ask you to find a suitable animal, and pay 10 gold in material components. The easiest way to get a companion is the Find Familiar spell. 10 gold in materials, and you get to summon an animal of your choice (1/4 CR), which get the fey, fiend or celestial type. It also gets the advantage that you can cast Touch-ranged spells through it, and you can see through its eyes. Can be dismissed and summoned as needed without needing more gold.

The second easiest way is to just work with your DM, and write it into your backstory. That's how my Dragonborn got his pseudodragon.