I've been going there for work and on holiday over the past 10 years, it's one of my favourite places.
To be honest the South and HCMC isn't my favourite. It does have a lively night life. The tunnels are just about worth a day trip, all a bit touristy though. There is a place you can shoot just about any piece of war time hardware. Have also spent a week at a beach resort Muh Nee, I hear there's a lot of good kite surfing there now, about 3 hours drive from HCMC.
My favourite is the Centre and North however.
Hoi An for beach and culture. If you've time and want something different take the train up to Ha Noi rather than fly.
Ha Noi great city, culture, museums (inc war museum as per HCMC), nightlife
Ha Long Bay - world heritage site, stay 3 days aboard a boat and venture in deeper than day trips
Sapa - night train from Ha Noi - mountains close to Chineese boarder, local ethic groos still in national dress, bit touristy these days. See if you can do a "home stay" to get away from crowds
Victoria Hotels Asia very good. Metrople in Ha Noi is best but other more interesting places in old quarter. agoda.com is good for Asian hotels.
As you have three weeks I'd strongly recommend 3 days 2 nights in Ankor Wat to see the temples (you fly to / stay in Siem Reep)
Drop me an email (in profile) if you've any other questions or want to see a few photos to give you a flavour