Hi Nicole !
First of all don't be scared ... it's going to be a wonderful trip ... people are so friendly that you'll want more ! Although Japanese people don't really speak English, it's really easy to find what you're looking for in a city like Tokyo.
Landing in Haneda is (according to me) the best airport since it's so close to Tokyo.
I agree with Mr Kirkland... best way to get to Shibuya...
As for the tickets, you don't have to actually buy a card but it's easier to move around since you don't have to worry about the price of the ticket.
If you don't want to buy a card right away, you can always buy a ticket to Shibuya and it should cost around 500-600 yens.
Also, in Haneda, just after the customs, you'll find a tourist information .. they're wonderful and will help you if you need anything.
Be careful as Narita is further away from Tokyo ... it takes longer and it's also much more expensive to go there. I've never left from there but I think it takes around 1hour and a half to get there...
As for the rest, I would absolutely recommend you to try to stay longer in Kyoto while you're in Japan... Tokyo's great, Osaka's ok but Kyoto is one of my favorites.
I wouldn't spend a week in Osaka ... 2-3 days are plenty enough (if you ask me) and Kyoto is SO different from Tokyo... you also have the Golden Pavilion and so many old and great places such as the Manda museum (worth a look if you know a lot about mangas)
You can also go to Himeji for a day, Nara, etc...but of course it depends on what you want to do.
As for Pokemon centers: Ikkebukuro is where you want to go (big big one), Sky tree has a new one, Tokyo Station (underground) has one and of course Osaka. There's also a new one in Kyoto... what's great is that you can actually find some exclusives from each of the centers.