My boyfriend was asleep and I had the urge to go through his phone
Why would you betray your boyfriend's privacy like that?
I want to know more about our relationship, and if I'm going to be in it for the long run.
You've already breached his trust and ruined your relationship. It is a huge violation of privacy and trust to go snooping around his phone. You are already telling him you don't trust him, and that will cause lots of fights in the future.
The relationship is not even serious, it wasn't anything too bad.
You say it wasn't serious, but you are concerned if you're in it for the long run or not. If you're having these trust issues then maybe you should actually end the relationship. This may be the best for both of you.