Newcastle Airport website has a really useful flight guide that tells you what flights are available and which airlines:
flight-guide.pdf
Also think about what temperature you'd like - some European city break destinations are very hot in summer, so not nice for walking around, but if you're going alone and not tied to school holidays, early May is a nice time to travel and often cheaper than June to September.
You say you have a tight budget, but what is this? Some people say this and then come up with a huge sum, which is twice what I normally spend on holidays without limiting costs.
However, the Nordic countries are very expensive when you get there, so needs to be considered. How about Gdansk in Poland? Cheaper than the UK, beautiful city, lots to see and do, easy train transfer from the airport to the city and you can also get the train to the coast, which is nice for a wander, even if you don't lay on the beach - probably won't be warm enough in May anyway.
Also think about what type of accommodation you want. I prefer self catering so I can eat out but also buy some provisions like drinks, snacks and breakfast stuff from the supermarket. If I'm going somewhere warm, I always make sure there's outside space and if I'm on my own, I prefer to eat out at lunchtime and then just have wine and something light in the evening on the balcony/terrace.