Did people used to travel by train before the introduction of cars?

Yes, and some people still do.

Rail, horses, buggies, wagons,  coaches, velocipedes (bicycles), ships, boats. balloons; Walking.
  Many still use trains, subways,  and ELs, every day..
  Trains are the Best to see a country. Or,   by airship.
    Edit: Trains are Hardly" dead", there is Amtrak,  Siberian Railway. Most of the world Outside of the  US still routinely takes trains.  Most of  freight still moved by rail. A huge marshalling yard in center of my city, serves Mexico, too.
  Cars were popular even with working class as far back as1920's. Model "T" only $800. My grandpa, a dirt-poor wheat farmer, had two of them  and 2 Fordson tractors. They really caught  on in big  cities during and just after  the Great War (WWI, now), they eliminated   refuse from horses. 

They did in the UK,  Europe and in USA/Canada.   About the same time in other countries.

It was the introduction of cars that caused the demise of passenger traffic on trains. More so in the US then many other nations.
So, yes they traveled by train if it was any great distance and they were not relocating. If relocating and they did not have a bunch of stuff to take along, the train was also the transportation of choice.
How ever not every city had a rail road connection, but often a rig (horse and buggy) could be rented at a local livery stable.
A side note:
A lot of cities elected to build paved roads for cars instead of investing in street car systems to reduce the costs of expansion of the city. Maybe if more cities had continued to develop mass transit systems at the turn of the 20th century things would be different today in many ways.

Before cars became common the train was the major way people did trips in the USA.   There were train tracks all over the USA not just crossing the USA but all over each state.    What killed the trains was cars,  the interstates and federal subsidies to the airline industry. 

EVERYTHING moved by train before there were cars.
And it's STILL the cheapest way to ship and travel so it's high time to eliminate trucks from our roads and go back to trains.
Trains are the greenest method of transportation by far.

Horses, horse and carriage, bicycles and trains were normal transportation and even after the car was invented it was the preserve of the rich until after WW2. Most people used public transport such as trams and buses here in the UK.