Answer
No, the railway should not be relied on as the main tourist transport network in Uruguay. Passenger service is limited to individual routes and does not connect most popular cities as intercity buses do. For routes to Colonia, Punta del Este, Piriapolis and other tourist destinations, it is more practical to look into bus or car travel right away.
Even if a suitable train service exists, you need to check the station, operating days and the last connection to the final destination separately. The schedule may be designed for commuter travel rather than for a convenient connection for a tourist travelling with luggage.
A train can be chosen as an experience in its own right when its route fits the plan and there is some extra time. But for a complex itinerary around the country, it is more reliable to plan around buses and add a rail segment only after confirming the operator's current timetable.