The university sport center was the main place for training and sport activities. They have many different acitivities such as yoga, volleyball, basketball and a regular gym. It was also very cheap and only ca 10 min by bike from LUMC.
As for social activities it was easy to come into contact with other exchange students through the organisation ISN which I recommend joining. They had parties and events, such as joint travels to nearby cities. Leiden united was a little bit more expensive. I definitely recommend it though. Through Leiden united you could visit fraternities in Leiden and eat food, which usually was pretty good. The Idea behind it is that you meet with 50 % dutch students and 50 % international students, so it aids in integration.
As for travels, the Netherlands are a good starting point but as far as I was concerned I did not enjoy travelling around the country very much. The nature is non existant, if you want to go for a run you will have to run on bike paths and there highest peak is a meek 300 meters. However, if you enjoy picturesque european cities it will not disappoint. For weekend trips I would recommend traveling around other countries as well. Belgium is only 3 hours away by car and France ca 5.