Is it true that you can never believe a man's word until he swears in the name of Allah? For example, in my case I should not believe this guy really cares for me unless he swears in Allah's name.I've always was taught that it's not good to always swear in the name of Allah, is that true?

Praise be to Allah, the Lord of the Worlds; and blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.
Sharia recommends us to have good confidence in Muslims judging their opinions favorably, and tells us to believe in what they say without asking them to swear unless the person talking to us is known to tell lies.
We are not required to ask Muslims to swear by Allah if they want us to believe them. But if they swear by Allah this makes us more convinced of what they have said. The Prophet said: "Whoever swears by Allah should be believed". [Ibn Majah]

Allah knows best.