Every substantial town has a kantor pos (post office). In tourist centres, there are also postal agencies, which provide postal services and are often open long hours. Sending postcards and normal-sized letters (ie under 20g) by airmail is cheap, but not very fast.
From Bali, mail delivery takes two weeks to the US and Australia, and three weeks to the UK and the rest of Europe.
Most post offices will properly wrap parcels over 20g for shipping for a small fee. Don't use the post for anything you'd miss.
International express companies like DHL, Fedex and UPS operate on Bali and offer reliable, fast and expensive service.