Hi, the correct phrases are:
stay at home = zu Hause bleiben
go to the sea = ans Meer gehen
go to the lake = An den See gehen
go to the mountains = in die Berge gehen
go to the cinema = ins Kino gehen
stay in the city = in der Stadt bleiben
go to the beach = zum Strand gehen
stay at the beach = am Strand bleiben
have a walk around the city = spaziergehen durch die Stadt spazieren gehen
go to the bar = zur Kneipe gehen
go the restaurant = ins Restaurant gehen
eat at the restaurant = im Restaurant essen
So they were all very close. In german you just put the verb at the end of the sentence. :)
Good luck for the presentation patty!