In the cash receipts method of accounting, sales are recorded when payment is received, and expenses are recorded when paid.
In the accrual method, sales are recorded when the transaction is completed. If payment is not received, an accounts receivable is made and the sale is still recorded. Same for expenses. When the transaction is completed, you deduct the expense, even if not paid. A liability is created in accounts payable.
In regards to your inventory, you are correct. If this is the first year of the business, opening inventory is zero, record the purchases of inventory, and then subtract the ending inventory to arrive at cost of sales.
If your sales are related to your classes, they can all be on the same Sch C. Gross receipts will be combined, subtract cost of sales, and you arrive at gross profit.