When you enter Vietnam unless you have a specific contract that gives you roaming at a discount you can end up paying 5Euro/USD per minute on receiving AND making phone calls to home.
The solutions is easy, buy a local card, but you need an unlocked phone GSM compatible and swapping SIM Cards nobody can call you for an emergency.
So unless it is for a very short stay, taking with you a second old and cheap phone seems the most practical solution.
Nearly every shops in Vietnam sell preactivated GSM SIM Card, the most popular are the standard Mobifone card, the biggest wireless provider and the one not spamming you with SMS every hour.
The normal voice and SMS SIM Card from Mobifone costs around 40.000 VND and is valid for 1 month and 7/10 days. After your first call you will see the actual expiration date, usually you gain 2 or more weeks. You also get as a bonus 50.000 VND for making calls. The card is sold naked, no instruction, the web site has more info and there is a number 9393 for English help. Pressing the sequence *101# the phone will receive info on credit and expiration date. Internationatl calls to USA and Europe are cheap, around 4.000 VND per minute, direct dial.
To extend for 1 month the expiration date you need to put 100.000 VND on the card usually buying a scratch card at any shops. If you just use the number for receiving calls and at expiration time there is still money on the phone it may be cheaper to just dump the card and buy another one. You have to decide if keeping the number has any value for you or not.
Using a second phone, Internet connection is most likely a non issue, every place has free wifi anyway, but there is also an Internet enabled Mobifone card with calls capability, see the web site for the limits.
In Vietnam when calling from your mobile to telephone, you always have to dial the city code (eg: 08 for HCMC) and the telephone number.
Here is list of Vietnam Area codes