I'm getting braces to close up gaps between small teeth- how fast do they usually start closing?

Braces. Average time 3-4 months but there are so many factors that can make this faster or slower such as extractions, rotated teeth, your age, and the list goes on.
Immediately. The moment you apply force to teeth they begin to move. Depending on location, severity, tissue health, possible presence of frenums, etc, movement could take weeks to months. No 2 people respond at exactly the same rate. Ask your qualified orthodontic specialist. He/she should be able to give you an individualized guesstimate.
Pretty soon. Based on your condition you can start seeing changes / spaces closing as soon as the first few months. It might take longer to complete the entire treatment but initial changes can be appreciated soon, which makes the entire process worthwhile and give you lot of things to talk about with your friends and family.