I can think of a couple offhand --
Aguri no Oka (Japanese site: http://www1.city.nagasaki.nagasaki.jp/ikoi/top_aguri.ht...) is a nice park in the hills outside of the city itself. It covers a big area, and aside from plenty of room to run around there's a playground as well as an open-air building with activites for kids (making tops out of chestnuts, little things like that) and I think places to sit and eat. Plus really nice views over the ocean. The easiest way to get there is probably by city bus. The stop is named, easily enough, Aguri no Oka (アグリの丘). Are you just visiting or staying in Nagasaki long term? I don't want to make assumptions. :) If you're already familiar with the buses, great, or if you speak Japanese check out the city bus website (http://www.nagasaki-bus.co.jp/), if not let me know. :)
Also outside of Nagasaki city itself is the Huis Ten Bosch theme park, which even has an English website that does all the explaining for me: http://english.huistenbosch.co.jp/ . I've never been there myself, though, so I'm not entirely sure what it's like.
If you're in Nagasaki long term there are some fun festivals coming up -- especially the kite festival and dragon boat festival -- all of which are listed here: http://www.nagasaki-tabinet.com/mlang/english/guide/eve... . If not, I'm afraid the best there is is a tulip festival at Huis Ten Bosch. ;)
Um... the Goto Islands have nice beaches? Problem is they're either expensive or slow to get to, depending on your method of transportation, perhaps a little much for a day outing. I wish I knew of a good train museum! Maybe try the tourist information office for more suggestions? It's right outside the main ticket gates of Nagasaki Station, and I think it was next to a coffee shop? It's a small area, so it shouldn't be too hard to find.
Hope this helped a bit. Good luck!