This can be a tricky one and will require a rather large vocab. base for success.
Luckily, many cell shops have English speaking employees that are keen on using their English on you... Let's not give them the satisfaction:

Them: いらっしゃいませ。
Irrashai mase
Welcome.

You：あの、携帯電話を申し込みたいですけど。
Ano, keitai denwa o moushikomitai desukedo.
Well, I would like to sign up for a cell phone.

Them： かしこまりました。お客様,今日は （1 or 2 or 3） をおもちでしょうか?
Kashikomarimashita. Okyakusama, kyo wa （1 or 2 or 3） o omochi deshoka?
OK. Have you (customer) brought （1 or 2 or 3） today.

1．外国人登録証明書
1. Gaikokujin touroku shoume sho
1. Alien Registration Card

2．銀行通帳
2. Ginkou tsuchou
2. Bank Book

3．はんこ･印鑑
3. Hanko/Inkan
3. Official Stamp

You：どうぞ
Dozo
Here you are.

Them:では,お好きな機種を選んでから、これをご記入ください。
Dewa, osuki na kishu o eranndekara, kore o gokinyu kudasai.
Then, please fill this out after you have found the model you like.

You： はい。じゃあ、これおねがいします。
hai jya,kore onegaishimasu.
Ok, well, this please.

Them： オプショナル留守番サービスをつけますか？
Opshonaru rusu ban sabisu o tsukemasuka?
Would you like to add the optional answering service?

You：あとでつけてもいいですか？
Ato de tsuketemoii desuka?
Can I add it later on?

Them： いいですよ
Ii desuyo
It's ok

You： Cool

When getting a cell phone, the general process goes as follows:
1. Go in the shop and look for the phone you like.

2. Fill out the necessary forms for that phone
(make SURE you bring the
A foreign registration card,
B your bank book (the phone bill will automatically be deducted from it), and
C the official stamp (hanko/inkan) you used when you made the bank account.

3. Pay for the phone (if there is an initial fee).

4. Wait 30-60 minutes for the staff to set up the number for you.

5. Take your home phone and set up the mail account.
You will usually get charge a service fee (for setting the phone up) included in your first bill.

Happy shopping!