There's a direct train from Keikyu Shinagawa station in Tokyo to Yokosuka . It may be best to get the express train to Kanazawa Bunko and transfer to a local train from there.
The best station for Mikasa park is Shioiri Station.
If you need to determine how much Yen to bring or to help you navigate the trains, this website is absolutely fantastic:
It will tell you which train to take, which station to leave from, where to transfer and how much fare. I suggest purchasing the cheapest ticket and then using the fare adjustment machines.
I love the trains here, but avoid them during rush hour!!!! I've been sardined into them several times and it's NOT COOL!