Newcastle is an excellent base for day trips. I will try to give you some insider information here.
BEACHES. The best beaches are in Northumberland. Getting there involves a bus trip from the Haymarket bus station (beside Haymarket Metro). My favourite beach is CRESSWELL. It's really sandy, with lovely big high dunes. It can be reached by bus X33 from Newcastle. Sunday and evening service is limited, so check carefully first. NEWBIGGIN has quite a good beach, and is a small former fishing village. It is quite faded and run down now. Try bus 432. SEAHOUSES is very popular, also as a destination for cruises to the Farne Islands. There is a 2 hourly bus service.
If you want convenience by public transport, go to those accessible by Metro. For all of these destinations, the Metro runs frequently from early morning to late evening. WHITELY BAY is a favourite one. The town centre has a good selection of restuarants and St. Mary's lighthouse is a popular visitors' spot. At the end of the day, it becomes a popular evening spot, especially among the younger population, with several bars. Check the times of the last Metro carefully! Nearby TYNEMOUTH is more upmarket, but suffers delusions of grandeur. It has some nice shops, and a good stretch of beach, but its sense of superiority is perhaps not justified. Still,the Tynemouth priory is nice to visit. SOUTH SHIELDS is another good choice. The town centre has a nice art gallery, and there are several good eating and shopping options as well.
POPULAR SIGHTS - The village of WARKWORTH is very touristy, but very very pretty. Warkworth castle is in ruins, but intact enough to give you a good idea of what it was like. In the village, St. Lawrence's church carries a lot of history. Warkworth was the scene of many former battles between the Scots and the English. Look out for Bridge End house by the bridge. This is a wonderful building, but sadly not open to the public. Bus 518 stops at the castle and in the village centre from Haymarket bus station. Nearby AMBLE is a short stroll down the river. They describe themselves as "Britain's friendliest port." That is a bit of truth-stretching though. I went to school there: I know. The marina is a popular strolling spot. Also in North Northumberland, BAMBURGH castle is finely preserved and is justifiably popular as a tourist site. There is a good beach there too. Ask at the Haymarket about the infrequent buses.
Going the other way, DURHAM is a historic city with a famous university. The Cathedral and castle are wonderful to visit. There is so much there that a prolonged stay would be better, but if you're doing it as a day-trip, trains leave from Central station often, and there are frequent buses. BEAMISH is an open air musuem near the town of Chester-le-street. It is set around a mining village. The drift mine tour is interesting and so are the tram rides. It is good for souveniour shopping and will provide a few hours entertainment.
OUTDOORS - Here you might be pushed using public transport. The Cheviots are fantastic for walking. The village of Longframlington is a good base and can be reached by bus. However, the buses to here don't run often and you are still a bit out. Rothbury is an old market town that provides some walking opportunities. It is reached by the same limited bus service as the above, but provides better accommodation options. The only budget spot on the scene is the camping ground though. My favourite outdoor spot is Harewood Forest. This provides several hiking trails in a thick, dense forest with several deer and other good wildlife spotting opportunities.
HADRIANS WALL is famous. This is better if you don't have a car. Bus AD122 from Eldon Square bus station runs along all of the main sites, including Vindolanda and Housesteads, and it calls at all of the main towns on the way, such as Corbridge, Hexham and Haltwhistle. There are some good hiking routes between bus stops.
KEILDER reservoir and forest is also a popular outdoor spot with several campsites. If you're using public transport, it will require an overnight stay. A bus goes from Newcastle about twice a week. There is a youth hostel nearby.