I would like to know the best things to see, the ones to avoid considering I'm focusing on all stuff related to the nature, hiking and off established trails? Thanks a lot !
If you're focusing on nature-related activities for your travels through Vietnam, you absolutely can't miss Ha Long Bay in the North (http://www.travelfish.org/location/vietnam/hanoi_and_surrounds/quang_ninh/ha_long_bay). (You'll have to get a glimpse of some photos to get a feel but it's truly one of the most majestic places you will ever visit.) With what seems like thousands of small limestone islands spread throughout a serene bay, the possibilities for outdoor activities is endless in this UNESCO World Heritage site. There is world class rock climbing, kayaking and trekking.
Should you require some local support, I'd recommend going with an outfit formerly known as Slopony (www.asiaoutdoors.com.vn). Also, you can rent a spot on a boat and spend a few nights drifting through the bay and taking it all in.
It sounds, Julien, like you'd benefit from focusing your trip throughout Central and Nothern Vietnam, where even the more metropolitan areas (i.e. Hanoi) retain some of the colonial charm. (While there are some interesting sites to see in Ho Chi Minh City, it's very much a major metropolitan, urban atmosphere.) There are lots of areas to go trekking as well.
Vietnam is fairly small, geographically speaking (at least compared to the United States) and three weeks will give you plenty of time to roam around these regions.
Mai Chau is impressively green, with rice paddies, palm trees and wonderful bamboo pole houses that you can sleep in. Lovely place to hike.
The Mekong Delta boasts great food, lovely people and the delta is a wonderful place for some cycling.
In terms of nature I'd also definitely recommend
Sa Pa. The views there are stunning and it offers great hiking opportunities, though it does rain there quite a bit.