I have done a wee bit of searching online, and I am indeed surprised. I see that the line from Dundee to Newtyle was opened in 1831, and the line from Dundee to Arbroath piecemeal between 1838 and 1840. The two lines had different guages so there could not have been any through trains.
So your folks might have gone from Arbroath to Dundee, and then from Dundee to Newtyle, but that would be a complicated way to do it, and how would they have continued their journey from Newtyle? Far simpler to go by sea and canal.