In Chester, the pair opened the Storyhouse, a library, theatre and cinema complex.
The two royals were greeted by cheering crowds in the centre of the city of Chester, where they observed the minute of silence.
Meghan is expected to curtsy to all royals of a higher rank to her, which includes the Queen, the Duchess of Cornwall and the Duchess of Cambridge. Meghan, 36, wore her hair down on the windy day and paired the look with black heels and a black clutch bag.
The Queen has her own train she uses to get around the United Kingdom for special engagements.
Protocol dictates that royals must curtsy in front of the queen when they see her for the first time of the day.
Britain's newest royal joined the monarch on a visit to the northwest county of Cheshire, where they opened a new toll bridge over the River Mersey.More news: Strong deadly quake rattles Japan
Ingrid Seward, of Majesty magazine, said when formalities are over, Meghan can call the queen something else.
Meghan Markle's new life as a royal comes with more than it's fair share of public scrutiny.
They wore contrasting colours, the duchess choosing a beige Givenchy pencil dress and the queen in a lime green coat-dress and matching hat.
Thousands of people are already waiting at the venues to try and get a view of the pair.
The locomotive travels much slower than most modern trains and operators even make sure that carriages do not cross any bumps while the Queen takes her bath around 7:30 AM.
For his next release, the Duchess chose a light Sussex slinky MIDI dress with a black small belt at the waist from the Givenchy. The unveiling of the 1.86 billion pound project was followed by a performance from local school children.