I'm sorry to hear that your child is having feelings of anxiety in first grade, especially at the beginning of the school year. It is important to get to the cause of the problem, so here are a few suggestions.
Talk to your child. I'm sure you have already done this, but ask specific questions (not just how was school today?) Ask about friends or partners? Ask who she gets along with or who she/he doesn't like and why? Ask about teachers, lunch ladies, office staff or anyone she comes in contact with.
Volunteer in your child's class. See for yourself what is going on.
Keep a journal. Joint down a few facts about what you observe daily. Record if your child is fearful about going to school, or when he/she comes home from school. When you talk about the day, what are some of your discussions. Notice body language when your child is talking. You are looking to see if there is a pattern. For example, do the anxious feelings appear on a certain day or when your child sees a certain teacher? Does it happen at a certain time during the day such as at recess?
Set up a meeting with your child's teacher. Share your concerns and finds with the teacher. She/he might be able to shed some light on what's going on. Ask if she has observed any academic weaknesses or other behaviors that may raise a red flag.
Talk to other parents about how their child feels about school. This is not a time for gossip, but instead stick to the facts. Ask them if their child has any problems or anxiety.