"The right side. Then the right side." الأيمن فالأيمن
The prophet said these words (just 2 words in Arabic) when he was sitting with an unknown Bedouin on his right side and his best friend Abu Bakr on his left. After drinking from the milk that was offered to him, the prophet passed it on to the Bedouin even though Umar suggested giving it to Abu Bakr first.
There are other Hadiths supporting this general preference of the right side. For example, we learn from those Hadiths that the right foot should go first when putting our shoes on and the left foot first when taking them off. In ritual purification we should wash the right side of the body first. We should enter the masjid (mosque) with the right foot first and exit with the left foot first. Unlike shaytan, the prophet used to eat, drink, give and receive with the right hand and commanded his followers to do the same.
One exception to this general rule is the case of using the bathroom where the left foot should enter first and exit last and the cleaning is done with the left hand. Activities such as blowing the nose and removing filth should be done with the left hand as well.
We should be careful not to let our arrogance be the reason we don't follow this or any of the prophet's sunnah. There's a Hadith about a person whose right hand became paralyzed because arrogance made him refuse to eat with it even after the prophet asked him to do so.
The Hadith teaches us that Islam is so comprehensive it addresses little details of our lives. It also teaches us that even our effortless daily activities such as shaking hands, lying down, or getting dressed could give us rewards for following the prophet's sunnah as long as we have a sincere intention and we are conscious of what we're doing.