These are the places I've been to and found very nice in Vietnam.
Sapa: a small town in the northwest of Vietnam with amazing mountain scenes, terrace rice fields, and villages of different ethnic groups. Tourism services are quite developed too.
Vietnam political and cultural capital. If you enjoy Vietnamese food, Hanoi has a lot to offer.
Tháp rùa (Turtle Tower), in the middle of Hồ gươm (Sword Lake) an iconic location in Hanoi.
Phố cổ (Old quarter) in Hanoi. One of the best places in Vietnam for food and observing Vietnamese culture. The old houses may not be well preserved but the Old quarter is a very lively area.
Mausoleum of Ho Chi Minh (founding president of Communist Vietnam).
Hồ tây (West Lake), the biggest lake in Ha Noi
Ninh Bình: Extremely beautiful but less well-known destination, about 80km south of Ha Noi. You'll enjoy an 1-hour ferry ride through a little stream and magnificent caves, surrounded by mountains and rice paddies.
Hoa Lu, the old capital of Vietnam (1000+ years ago) is in the area too. You'll find some temples of ancient Vietnamese kings here.
Temple of Đinh Tiên Hoàng, one of Vietnam's first kings.
Has many large and beatiful caves. Tip: Try the Thien Duong Cave, which is a lot better and well-preserved than the more well-known Phong Nha Cave.
Thien Duong Cave
Thien Duong Cave
Clean & beautiful beaches, very good tourism services.
If you ride a motorbike, don't forget to take a ride through the HaiVan pass. You'll see the beautiful mountain and sea from the above. This is also a historical border of Vietnam and the kingdom of Champa.
Hoi An: About 30km from Da Nang. Maybe the only well-preserved old town in Vietnam. One of the drawback is although the old houses are preserved, they're now loaded with commercial shops and hotels. Most people don't live in the old town anymore.
Close to Hoi An is Cửa Đại beach, a cute little beach.
Old political capital of Vietnam in the 19th century of the Nguyen dynasty. Here you'll find the remains of the dynasty palace and many kings tombs.
Gate to Đại Nội, the old palace.
Tomb of Khai Dinh, a Nguyen king.
3. South Vietnam
A mountainous region with temperate weather and many nice sort of flowers.
Tuyen Lam Lake
With sand dunes, beaches, and well developed services.
Ho Chi Minh City (Saigon)
The biggest city in Vietnam. Old capital of South Vietnam.
Independent palace, used to be a resident of South Vietnam presidents.
Notra Dame - built by Frenchmen in 19th century.