I love Japanese trains. But I really wouldn't bother riding the Shinkansen for the short distances you mentioned. They won't get up to full speed (neither is either of them anything near to leaving the Tokyo metropolitan area), and I fear you will be very dissapointed. There is nothing to see in Shin-Yokohama. Omiya itself is also totally non-interesting, but you can go from there to the JR East Railway museum, which is quite good. It's always great fun to see people using the train simlulators with their own train driver's 'uniform' and kit.
The metro (both the Toei and Tokyo Metro systems) are very dull and workmanlike. Not for riding for fun. Definitely avoid in the morning rush hour or on the last train at night.
In my view, where you can get the most fun, is by riding some of the suburban systems. If travelling from Narita airport, ignore the special airport lines, and take the Keisei Main Line Limited Express, and get the Driver's eye view as you travel through the Chiba countryside and East Tokyo suburbs (for around one quarter of the price). From Haneda, ignore the overated and slow Monorail, and take the Keikyu. Again, you can sit at the the front, and see the trains hurtling along rather faster than feels safe on such a busy line. Depending on where you are staying in Tokyo, take the Yokosuka line or the Shonan-Shinjuku line to Kamakura, and get a full day of historic Japan in one day. If you want to go on some unusual trains, you can get the Enoden tram, and the Shonan Monorail (the suspended type) while you are there.
The JR Railpass never makes sense unless you are going to do some serious long-distance travelling within Japan.
If you are there for more than a few days during holiday season and will get out of Tokyo, I reccomend getting the Seishun 18 ticket. It is the hardcore way to see Japan by train. If just a few days in Tokyo, I would try to do the following:
- Harajuku and Yoyogi Koen on a Saturday afternoon to see the various 'tribes' (if you are really fortunate, you will see the Rockers who have to be seen to be believed), and spending some time to see Meiji Jingu (one of the most famous shrines in Japan)
- If you are curious about how the revisionist side of Japan sees the second world world, go around Yasukuni Jinja (but try to hide your horror at some of the claims you'll see made)
- Hachiko Crossing in Shibuya in the evening
- Go up the Tokyo Metropolitan Government Building near Shinjuku for one of the best views of the city (and on a clear day, Mount Fujii)
- Walk around the crazy malls of Odaiba (the other side of the Rainbow Bridge, and it's decidedly odd yurikamome line)
- Asakusa, and Akihabara, if you want to go to geek central (but watch out for pickpockets)
- Some enjoy the Yamanoto Line walk. Pretty exhasuting, but you get to see most of the interesting sights
All of the above are free (once you get there).
If there is a tournament on, it is definitely worth going to see the Sumo. Buy a standby ticket in the morning, and actually go in in the late afternoon when the real heavyweights come out. Watching the (normally very drunk) crowd is even better than the wrestlers.
Download the Japan Wifi app (from NTT) before you go - it will get you free Wifi in many places. Get to as many Ramen bars as you can (and don't pay more than about 600 yen), and conveyer sushi places (don't pay more than around 200 yen per plate). Avoid hotel resturants. Go into 100 yen shops, but think about what you are buying. Don Quihote in Shibuya is a great place to buy souvenirs and weird Japanese stuff. Don't buy electronics. If you drink alcohol, Sake is super cheap (250 yen for a 300ml jar of 'One Cup' is very effective), and Chuhai is like alcopop but more sohphisticaed, cheaper and normally stronger. Avoid Roppongi and Kabukicho like the plague, unless you want to be ripped off or catch something nasty (when people say they are 'popular with tourists', they mean a certain type of tourist). Any bar or resturant with someone outside trying to entice you in is better avoided unless you speak Japanese. Karaoke is great if you read Japanese - throughly miserable if you don't. The Disney resorts are great if that's your bag but a bit of a waste of time if it is limited.