The town of Hekou is within walking distance of the border post, and Lao Cai is just across the bridge. The border is open 8AM-11PM (7AM-10PM Vietnamese time).
Trains link Hekou North Train Station (河口北站) with Kunming twice a day over the new railway opened in December 2014. Trains leave Kunming for Hekou at 9:28, 12:45, 16:00, and 22:48. The trip takes six to seven hours, and costs RMB 70 for a hard seat and RMB 160 for a soft seat.
Buses run from various destinations within Yunnan, including Kunming (¥140, 6 hours, 9:40, 10:40, 12:40, 19:00, 19:30, 20:00). Some connections, including Jinghong require you to change buses in Mengzi. Bear in mind that the bus may not take you all the way into Hekou, but might stop at a point West of town. It's about an hour's walk from here to the border post, along the Honghe river. Or, you could take a taxi (there are usually some waiting) for around ¥10.
When you leave the border office, ask the Chinese officials for a small guide with basic map of Hekou map and a glossary, if it is available.
Beware of helpful guys speaking English at the bus station, offer to sell bus/train tickets in Vietnam for triple their real price or exchange currency several times more expensive than in respectable places. Check prices and exchange rates in advance. Tickets can be bought easily at Vietnam side of border.