Ho Chi Minh City is divided into 24 districts: 7 named urban districts, 7 numbered districts, and 5 outlying named suburban districts. Most of the things to do in the heart of the city occur in District 1 and District 3: this is where you'll find most of your sights, museums, best eateries, bars, etc. For this reason Districts 1 and 3 are the best areas to stay in as it can be time consuming and energy draining catching taxis all the time back and forth from your hotel if you are staying too far away from the centre of the action.

The best neighborhood for the budget traveler to stay is Pham Ngu Lao, where cheap but reliable guesthouses abound. You'll find the least expensive accommodations tucked just a few steps away from the most obvious options, tucked away on the alleys (this goes for throughout HCMC). You can find your full of food here for USD $0.35 and up--don't leave Vietnam without a hearty dose of cheap street-side pho' (a hearty noodle soup usually eaten for breakfast).

The Dong Khoi area is one step up in terms of ammenities and prices. While you're in Ho Chi Minh City, be sure to check out Cho Lon ("Big Market"), where the city's ethnic Chinese population is concentrated.  The market is also known as "Cho Tan Dinh" where you get fair prices although be aware of "fresh" quotes. Make sure you negotiate pricing for 20-30 % off the quoted price.   If you are a foreigner, tell them that you are an expat and are living here, so please give you a fair price.  Learn some VN will earn some respect and invite a more friendly negotiation experience.  

Another large market is just a few blocks away from the five-star hotel, Sheraton Saigon.  It is a wise bargainer who shops somes of the hotel shops, small stores before venturing into the markets.   Another tip - bargain in Dong, not USD as it is much more finite in pricing and the total price will be considerably less rather than rounding to the nearest USD (around 21,000 Dong to $1 USD).