Best way to travel to and inside Vietnam

Trains do not connect all places in Vietnam - there's effectively one line that runs from Hanoi to HCMC along the coast for most of the route, and a couple of branch lines (to Lao Cai and Hai Phong) in the north.

You can see a map here (note that the route to Halong appears to be a bus) Note that not all trains call at all stops:
_{(source: seat61.com)}

However, most of the places you mention are either accessible by train, or the train comes close to and you can transfer (Lao cai for Sapa, Da Nang for Hoi An).

However, Vietnamese trains are not very quick. For longer distances, flying is quicker and often comparable in price (although note that the cheaper Vietnamese carriers are notorious for cancelling underbooked flights). Bus services can be cheaper and competitive with the train on times as well (although considerably more dangerous).

The advantage that the trains can offer is overnight sleeper services, so that you can sleep on the train and still get a full day in at either end. Note that some people may find sleeping on the train much easier than others - these are not smooth European high-speed services - the trains sway considerably and will make stops during the night.

In my experience, the best way is by motorbike. Although buses might seem an easier choice, the danger and incompetence of the drivers actually make driving yourself on motorbike a lot safer and more efficient!

Given the tight schedule and that you only want to visit specific places, using a local airline with gaps filled in by bus is your best bet. Vietnam has some of the cheapest local flights. For example, earlier this year I got Hanoi to Danang for 35 Euros, all fees and taxes included.

The bus network is extensive and frequent. This can save you money version flying at the expense of time. Still, you can save money by choosing overnight buses on long segments which are surprisingly comfortable and spacious. While you can contact the bus station or get there to get info, I was always able to get a fair price and necessary information of English speaking staff hotels that would often call several companies to get schedule and pricing. Even taxi drivers and have a hard time with English, the hotel would tell them were is the pickup and write down to drop off for me to show them later.

When I travelled the length of Vietnam, I originally looked into train but after investigation found them slow. Plus you need to make your own connections while tourist bus service can pick you up and drop you off at your hotel if it is within a certain area. This easily saves a good amount. Same goes when comparing to flying. Transport from airports can be relatively expensive.