I wouldn't bother with the train into Laos since it only goes 3 km over the border and leaves you just as far from Vientiane as the bridge does.
The easiest way is to get a tuk-tuk from the train station to the bridge (about 1.5 km), cross the bridge etc and then get a taxi to Vientiane.
You can also wait and take bus number 14 or any of many shared tuk-tuks from the bridge to Vientiane if you want to save some money. The bus will take you to the bus station at Talat Sao (Morning Market) and the tuk-tuks will generally take you to the same place but will take you on for a bit extra.
Agree the last leg of the train over the bridge exists only for fanfare at this stage and saves you no real time/distance getting to Vientiane . In Nong Khai, after you deal with immigration on the Thai side you'll see a desk selling 20 baht bus tickets over the bridge (leaves every few minutes when full). Then do the visa/immigration routine on the Laos side, after which you'll see another desk selling 50 baht bus tickets into Vientiane (arrives at Talat Sao). There will also be taxi and songthaew (covered pickup truck or moto rickshaw) drivers offering rides into the city for anywhere between 50 baht per person to 400 baht for a car. The bus always suits me fine but they do pack them full.