Gedser is a small port town on the island of Falster in Denmark's Zealand region, with just 832 souls living there, it's main claim to fame is as the southernmost town in Denmark, and the ferry line to Germany.
The old train station in Gedser is served by a lone train per day departing Copenhagen in the morning, arriving 2 hours later at early noon and returning around 11AM. there are plans to shelve this route completely, but authorization for this has yet to be granted by parliament, hence the one train, but remember to check if the train is still running if you plan on taking it.
Movia  line 39 connects Gedser with the regions main train station in Nykøbing Falster around once per hour on weekdays, down to 8 departures on Sundays. In summer another option is line 43, also connecting with Nykøbing Falster, but stopping at many of the regions popular beaches en route.
The main reason to come to Gedser is the ferry to Rostock in Germany, the most direct route between Copenhagen and Berlin. The crossing takes 1 hour and 45 minutes and is operated by Scandlines . Departures every two hours between 7 and 23. Price is 650 DKK for a car including passengers, and 50 DKK for walking passengers.
Gedser is very much small enough to walk anywhere you would need to go.
There is a single gallery, and a grocery store in town, near the ferry terminal
Nope, but you can buy a beer at the grocery store and hang out at the harbour