You can visit many places in three weeks. If you will be Hanoi first, our capital, you must go to Sword Lake- heart of the city. You probably explore other well-known landscape in Hanoi such as Thang Long Water Puppet Theatre, Literature Temple, One Pillar Pagoda, West Lake, History Museum on foot or by cyclo, it is quite cheap. You will absolutly be attracted by Hanoi's cuisine such as "Phở" - Rice noodle soup, "Chả cá" - Grilled fish, "Bánh tôm" shrimp in batter and "Bánh cuốn"- stuffed pancake, so on. Then, you can go to Quang Ninh province (about 150 km from Hanoi) to see one of new seven wonder of the world - Halong bay. You should visit Ninh Binh province, 3 hours drive out of Hanoi, with many sights such Tam Cốc caves, being known as the inland Halong bay and Cúc Phương rainforest - home of rare species (but you will meet visitors mainly hahaha).
To go Ho Chi Minh city, you should go by train to go through many provinces in Vietnam.
Note: You should not stop and show interest in what some street sellers talk you into buying their goods or services. If not, you will buy will exorbitant prices, or you will be treated with no respect when refusing. In additions, you must always beware of lurchers when you are in crowd. However, Do not worry, Vietnamese is very friendly. You can talk to every one and ask for direction. Use your body language and always smile.