Can I charge a different shipping fee for each product?

Yes. Manual shipping (Settings → Store Settings → Shipping Setup) supports per-product shipping, so each product can carry its own shipping charge. It also offers an overall charge, and free shipping over a minimum order amount.

For rates that depend on where the customer is, use the Shipping Zone plugin. Couriers connected with the Shipping Carriers add-on create labels and tracking after the order; they don't set the checkout price.