You might want to start in Malaysia first. Most people speak English and it will give you are fair idea about cycling in that climate and region, while being a "safer" start.

In Vietnam, which is lovely BTW, not many people speak English. We had to ask the hotel staff or point to a map to tell the taxi driver where to go. The cities and traffic is chaotic- lots of motocyclists etc. There seems to be an unwritten rule, whoever honks loudest you give way to. Bikes are the bottom of the food chain and give way to everything. However out in the country it's great. Hue is a must see (the old capital where the empreror used to live), Hoi An the rustic riverside village outside of Da Nang and Ho Chi Minh city is even great. Lots of resturants, businesses and apparently more commerce orientated than Hanoi in the North.

There are heaps of cycle tour companies in Vietnam with and they will supply bikes- most likely mountain/hybrid road bikes. You can of course bring your own pedals and saddle and being a tour company they should be well maintained.

If you are thinking about bringing your own bike I am not sure how well buses or trains would go, and whether the bike would be handled well. That's why I think it would be better to rent from a tour company instead, Saves having to haul it around with you.

I haven't done a bike tour in Vietnam- we were only there for short stays, however I have cycled in Malaysia and had a great time and cycling is becoming big as a sport (and not just a mode of transport) with plenty of locals out on Pinarellos etc. They even had a Malasyian Navy Cycle team!