Tokyo Monorail offers a discount ticket between Haneda airport station and Shinjuku for weekends special offer of 500 Yen(normally charged 690 Yen). You can save a bit if you buy the ticket.
You can buy it from the ticket machine with your IC card. There is an IC card slot on the ticket machine.
Or ask one of Tokyo monorail staff there how to buy a weekends discount ticket with your IC card.
http://www.tokyo-monorail.co.jp/english/tickets/purchase.htmlBut you need to go to Hamamatsu-cho sta by Tokyo Monorail first, switch to Yamanote line for Shinjuku sta next. You must use the discount ticket as you go in Haneda airport sta and Mahahatsu-cho sta, it will automatically be collected in Shinjuku sta.
At Hamamatsu-cho sta, go to platform No,3, which the Yamanote line goes to Shinjuku.
For Yokohama,
By JR line, you can go by Shonan-Shinjuku line for Yokohama from Platform No.1.
By Tokyo metro
Go to "Shinjuku-sanchome station" which is about 7 mins walking distance from Tokyo metro Shinjuku sta.
Go to platform No.3 in Shinjuku-sanchome station. Fukutoshin line takes you to Yokohama station. Check the destination first, Motomachi-Chinatown destination takes you to Yokohama, but, "Kikuna" and "Musashi-Kosugi" destination stops before you get to Yokohama. You can wait for the other train which goes to Yokohama if the train stops in Kikuna or Musashi-Kosugi stations.
Your IC card balance can be checked with ticket machines. You can spent your balance at major supermarkets, combini stores, station kiosk and some station shops(restaurants, coffee shops, cake shops, souvenir shops etc. You can also ask to check the IC balance at those shops.