Can I see how many new customers I got each month?

Yes. Go to Analytics → Reports and open New customers over time (filter the list by Customers or type it into Search reports…).

Related reports in the same category:

• New customer sales over time: how much those new customers spent.
• New vs returning customers: the balance between first-time and repeat buyers.

For a simple count of accounts created in a period, you can also use Registered from and Registered to on Customers → Manage Customers; that list also includes accounts created by guest checkouts.