1. Could anyone explain me how to get to the Dinosaur Park from Seoul? What transport should I use? (bus, train? their numbers?).
I'll tell you how to get there considering you are in Gang-nam station.
1) take a subway Sin-bundang line신분당선 going to Pan-gyo station. the color of this line is crimson red. (about 20 mins)
2) transfer Pan-gyo station to Gyonggi-Gwangju line경기광주선, the blue color, going to I-cheon/Yeoju station. (about 45 mins)
3) get off in I-cheon station. and take the bus 22-4 going to jakchon-ri termination. (about 1 hour)
4) and then walk about 1km on foot. (about 15 mins)
cost : about 3 $ (if u take a taxi from I-cheon station, it could be approximately 20 $)
2. How long does it take to get there from Seoul?
From Gangnam station, it would take about 2 and half a hour.
However, the term of the bus 22-4 between the next bus is almost 6 hours and it's really hard for you to know the schedule of this bus. So, I'd like to recommend to take a taxi in I-cheon station rather than take a bus. the taxi cost would be 19,000 won(about 16 $) for one-sided.
3. And what is the weather like in Seoul in the beginning of May? :)
You'd better think it as mid-winter in Korea.
If you wanna know the course more, message me.