I have a six-year-old son who is in first grade. He really seems to enjoy school but I have concerns about how he is doing socially. He hasn't yet connected with any specific child who he refers to as his "best friend." He does have friends that will come over and play but no one that he is really close with. When asking him about his school day, it seems that some of his recess time is spend alone on the playground. This may be something that concerns me more than him since he has never said that this upsets him. My question is , should I even encourage my son to be doing more to initiate a closer friendship or should I leave it alone as long as it's not bothering him?
Kids this age do not need to have "best friends" to be developing socially. Your son appears to be able to play with kids at school and at home. He is not complaining about his social life and you have not observed anything worrisome about his behavior. Some kids need best friends, some get along just fine with several acquaintances, some like to come home after being with friends at school and hang out alone with their dog in the woods - it's all OK.
You can ask open-ended questions about his social life ("So whom are you playing with at recess these days? Do any kids in school like to [insert a hobby or interest of his]?") as you would about other aspects of his life. First grade is a big year in a child's life, full of novel experiences and feelings - stay connected with his world and he and you will be fine.