At what age do you think it's best to allow a son/daughter to enter relationship nowadays?

Late teens or later. It's important for young people to experience their childhood before they delve into more mature matters like romantic relationships. If they do decide to start early however, parents should be responsible for giving them all the necessary information and knowledge about relationships and romance.

After school... Med school. Haha kidding aside, I just want her to be on an age when she's not going to play around and she's mindful of the consequences of her actions.

If it's up to me, after college graduation. But I can't control his/her feelings. So I'll just guide her in case she'll enter a relationship earlier than what's ideal for me.

Depends how self-aware they are and how they are able to discipline themselves. Usually ready by age 17.