Saul is anointed with oil to signify that he was God's choice to be king. Saul was chosen as king for the people and he was the type of king that people wanted- tall and handsome. Two things that are significant things take place here.
Saul receives the Holy Spirit. As Christians we receive the Holy Spirit when we put our faith in Jesus. We can look to a specific event in Acts called Pentecost to see that initial coming of the Holy Spirit on the disciples. Some mistakenly think that the Holy Spirit only comes into the story there. Throughout the Old Testament there are occasions and people who were filled with Holy Spirit.
The second thing that I think is significant is that not everyone accepted Saul as King. He already had doubters that he would be a good king. Notice where these critics are found; in Gibeah, his hometown. Saul returns home after being anointed King and some in his hometown doubt and despise him. Was it jealousy? Was it maybe personal- Saul said this to me or did this to me? Perhaps it was just a matter of knowing him too well, knowing his weaknesses may have caused some doubts. All we know is that in his hometown he was met with criticism and doubt. This type of experience will be seen again in Jesus' life when he returns to his hometown and they doubt him.