This translation is intended for people who already know English and are learning Japanese. It is not intended to be used to learn English, because the sentences in English may use strange or awkward wording in order to illustrate the vocabulary and grammar of the original Japanese.
Intro
Japanese
フロリダビーチでバカンスを たのしんでいるタートルズ・・・
English
The turtles are enjoying a vacation at Florida Beach...
This is a case of taigendome (体言止め), ending a sentence with a noun to emphasize it.
TMNT
Japanese
「エイプリルのリポートが はじまるぞ！」
English
"April's report is starting!"
April O'Neil
Japanese
「ここニューヨークでは、はんざいが ふえるばかりで・・・・・」
English
"Here in New York, crime keeps rising, and..."
TMNT
Japanese
「な、なんだ！？」
English
"W-what's this?"
Shredder
Japanese
「タートルズよ！ みるがいい！！」
English
"Turtles! Pay attention!"
Japanese
「タートルズよ、エイプリルと マンハッタンは、あずかった。」
English
"Turtles, I have kidnapped April and Manhattan."
Japanese
「とりもどしたければ、おれのところへ、こい！！！」
English
If you want to get them back, come to my place!
取り戻したければ is the conditional form of 取り戻したい, which is the desiderative ("I want") form of 取り戻す.
The comma after へ is unnecessary (though the one after とりもどしたければ isn't unusual). It seems that this game prefers to use commas rather than spaces after phrases to make the all-kana text more readable.
Stage 1 - Beach
Japanese
レッツゴー！ タートルズ！！
English
Let's go, turtles!
Stage 2 - Ocean
Japanese
カワバンガ！ めざせマンハッタン！！
English
Cowabunga! Head for Manhattan!
TODO — Unusual word order
Stage 3 - Bridge
Japanese
あしもとに、ちゅういせよ！！
English
Watch your step!
せよ is one of the imperatives of する. More common is しろ; using せよ gives it more of an air of authority (like a military command).
Stage 4 - New York
Japanese
そらのうえの、マンハッタン！！
English
Manhattan up in the sky!
Stage 5 - Sewer
Japanese
ちかにせんにゅう、げすいどうでファイト！！
English
Let's sneak into the underground sewers and fight!
Stage 6 - Technodrome
Japanese
エイプリルを、さがせ！！
English
Look for April!
Rescued April
Japanese
「ありがとう、タートルズ！！」
English
"Thank you, turtles!"
Japanese
「マンハッタンとうの、じょうくうに、クラングのうちゅうせんが、あるわ。」
English
"Krang has a spaceship in the skies above Manhattan Island."
Japanese
「みて！！ シュレッダーがにげるわ！」
English
Look! Shredder's getting away!
Stage 7 - Building
Japanese
のがすか、シュレッダー！！
English
Are we really letting you escape, Shredder?
か, when used with the plain form, typically adds an incredulous tone.
Stage 8 - Spaceship
Japanese
さいごのたたかい、あいことばはカワバンガ！！
English
The final battle. The password is 'cowabunga'!
Secret message
This message can be accessed by entering the Konami Code on the title screen (up, up, down, down, left, right, left, right, b, a, start).
Japanese
このたびは、コナミのT.M.N.T.2 ザ・マンハッタン・プロジェクトをお買い上げいただきまして、まことに、ありがとうございます。
English
At this time we thank you greatly for buying Konami's TMNT 2: The Manhattan Project.
