Get to the middle of the country and spend some time in Hoi An - a UNESCO World Heritage site with centuries-old buildings. The small town has a very laid back atmosphere in stark contrast to the insanity of Hanoi and the other large cities. This is also where they make all the colored silk lanterns and the whole town is lit up at night. While there, rent a bike and take the quick ride to China Beach where you can rent a beach chair and umbrella and think about what it must have looked like 40 years ago.
Also, head to Hue. They are still re-building the palace (like a mini-replica of Beijing's Forbidden City) that was badly damaged during the war. You can take a tour down the river to see the old imperial tombs or rent a motorbike to do the same. Hue is also the starting point for most DMZ tours if you're interested in that. (If you do go on a DMZ tour, make sure you get to the tunnels - they're insane.)
If you want a more resort-y beach, the under-touristed Mui Ne (halfway between Hoi An and Saigon) has some great resorts (but not much culture). While there, get in an ATV and cruise over the red sand dunes.
From Hanoi, you can take a couple days and go up to Halong Bay as mentioned above, or take the night train into the mountains to Sapa near the Chinese border and do some trekking.
When planning your itinerary, bear in mind that the trains can be very slow. If you plan to traverse the whole country from Hanoi to Saigon by train/bus at a leisurely pace, plan on two weeks at an absolute minimum. And as mentioned, enjoy the cuisine! The Lonely Planet is pretty indispensable for Vietnam.
Edit: Just saw that you're going for 10 days. Unless you're really intent on seeing Saigon, I would try to fly into Hanoi, get down to Hue/Hoi An, then back to Hanoi and spend a couple days in Halong Bay or Sapa if you have time. The internal flight between Hanoi and Hue isn't too expensive and, if you have more money than time, could let you squeeze in an extra day of fun.
Last edited by Masterkova; 06-14-2012 at 12:21 PM.
Listen to "Fortunate Son" about 20 times on repeat before you go~
Try the pho for sure!! And watch this as a precursor...
Scooter~ ...and I guess you'd say I'm on my way to Burma Shave...
Oud. Buy some quality oud. Then send me some.
Seriously, have a great time. Sounds like an adventure.
The Definition of a Gentleman: Someone who is never rude, except on purpose.