"Each day I love you more."
Translation:Cada día yo te amo más.
From the perspective of a native speaker im pretty sure te quiero is in a lighter sense of love, usable for someone you like or family or friends. Te amo is more intense love (mostly just for partners i think). Both mean i love you (and i want you in the sense of te quiero) but context matters and they hold separate connotations.
With sentences concerning a verb and a living person (and even some animals) 'a' must always be used. Example: Yo las busco a ellas (mis hijas). Yo no lo busco a mi perro, Juan. This means that with direct pronouns you must always doubly clarify that people are the direct objects. Example. Yo las busco a ellas. Yo no los busco (other non-living things). HTH!