I dont think there are any game shop in the Lemvig area, other than the local bookstore which only got, if any, the most common of games.
However I can help you on 'the things to see and things to do' stuff.
Lemvig is one of the most beautiful areas in Denmark with wonderful view of the harbor and water from the hills surrounding the town.
You should visit:
Museum of religious art - This is a museum with international reputation and for the art interested, its worth a visit.
The Planet Path - A model of the solar system in which the planets and the sun are placed on the rute in the correct distance and size on a scale 1:1 billion. Visit the local tourist office in Lemvig to get leaflet with the route and such. Its a quite beautiful walk in the town.
The Sculpture Path - Another path, this one showing 54 sculptures.
Bovbjerg Lighthouse - A lighthouse that should be worth a visit. Close to the lighthouse is a bronze Age barrow as well.
Look at this website for more info on the above as well as some info on activities like golf and laser games you can participate in:
http://www.visitlemvig.dk/welcome.htm--
There are some locations close to Lemvig that are worth a visit:
JyllandsAkvariet This place is definitely worth a visit, especially for kids. The place can be found in Thyborön. A combi ticket can be bought there that also gives access to:
Kystcenteret Thyborön This is the largest exploratorium on the Danish North Sea coast. Its possible to experiment with waves, wind, currents, tides, quicksand, storm surges, dikes and so on.
The danish west coast was also fortified with bunkers during the second world war, and in Thyborön there is an old sea fortification on the beach that is worth a visit.
More info on the above, plus more can be found
here.
Hope that offers some help