It sounds to me that your local weekly crit is more of a training race type of thing. That isn't to say that it won't be crowded or anything unlike "real" racing.

What kind of group riding have you done? Before you start racing (even road races), please get in some fast group rides where you can get used to riding at race speeds in close quarters (even if there are slowdowns and regrouping points) and do some cornering practice. Cornering doesn't have to be done in a group for you to get techniques down and learn to be more comfortable at speed.

If the course at the local crit is open in between fields, then that is a good time to practice. Go one week as a spectator first but take some laps in between the races if you can to practice doing a course like that at speed on your own and with closed roads. Watch the races from the corners and pay attention to the line the racers are taking. Also note how much more smoothly the front of the field will go through than the back (if you're in the back, you will slow more before the turn and then have to catch up after).