Which should I learn first?

I am beginning to teach myself some basic Japanese to help me during my trip to Japan, and I am wondering which written language would be most useful for me to learn: Kanji, hiragana, or katakana?

all of the writing systems will serve you well. You need to know all of them. Kanji however, if you want to get by with limited Japanese knowledge, would give you the idea of what they're trying to say. However, kanji is also the writing form that takes the longest to comprehend and is no easy task.
For now, start out with hiragana and katakana, get used to them. Memorize what they look like and be able to read them. From there, you can start Kanji, as you should know how to write their Japanese sounds in hiragana/katakana.
What helps me learn kanji is talking. As I talk.. i try to visualize the kanji that go with my sentence. Sometimes I can do it, and sometimes not. But if I can, then I know I've memorized that kanji for good.

It depends a lot of what your intentions are: If you want to be able to communicate in Japanese during your vacations, it will be best to start with hiragana and katakana; it won't take you more that one or two weeks to learn those. You can get a long way in Japanese without knowing much kanji. However, on the other hand, if you are getting very serious about Japanese and you also want to be able to read well, you'll need to start learning kanji as early as possible.
I bought the White Rabbit kanji flashcards for my own studies: http://www.whiterabbitpress.us/

I think learning kana first is better and will help you in the long run, though I don't think it matters whether it's katakana or hiragana (I learned hiragana first). Kana is so much simpler than kanji and easy to read. It will also help you with reading kanji because there are a lot of kana (or kana-like) strokes in kanji. It will also help you get used to counting strokes and stroke orders. For me it has, at least. :) Good luck!