Neringa is not only the favorite day trip from Klaipėda but one of the most favorite destinations in Lithuania. This string of fisherman villages that became clean prestigious resorts boasts spectacular nature and many opportunities for calm tourism. You will need to use a ferry to Smiltynė and then a bus that traverses entire Lithuanian zone of the narrow peninsula. Juodkrantė is 21 km away from central Klaipėda, while Nida at the southern limit of Neringa is 50 km away. If you have your doubts, let me say that many cruise ship passengers opt to spend their Lithuanian day in Neringa and therefore skip Klaipėda altogether.
Another possible day trip is the Palanga resort, 26 km to the north (highway, frequent bus services). This is the resort to choose for crowds, night entertainment and a flavor of kitsch. That said, there is much to see in Palanga including a large amber museum inside a 19th-century palace that is surrounded by probably the best park in Lithuania.
A trip to the south will reveal you the mainland Lithuania Minor. The largest town here is Šilutė (54 km away). Near Šilutė is the Nemunas Delta Regional park, a great location for birdwatching, angling and boat tourism (at costs lower than in Neringa). There you may find Rusnė with its landscape flooded by Nemunas every spring (61 km to Klaipėda). Even closer is the village of Minija / Mingė (51 km) that is nicknamed "Venice of Lithuania" because the building facades here face the unbridged River Minija and people use boats to visit their neighbors on the opposite side. All three and the Ventė horn ornithology station may be easily visited on a single day trip, but you will likely need a car.
A somewhat longer drive away from the shore to Samogitia National Park (~75 km) will allow you to descend into a shaft where Soviet nuclear missiles once waited to be launched (now a Cold War museum). Nice manors of Kretinga and Plungė may be visited en-route.
For shorter and more casual family trips one may choose a dinopark in Radailiai with its moving sculptures of pre-historic animals and a mini-zoo in Jonušai which is not really that small with tigers, kangaroos, warthogs and other animals. Both are ~10 km from Klaipėda. Another positive trip is to the "Nature pearl" park north of Klaipėda which has some live dear but its main draw is the repository of various exotic hunting trophies of a local businessman.