On the airplane there: be courteous to the other passengers and the flight crew. Pray in your seat, standing only when the fasten-seatbelt light is turned off!
Some broad outlines:
- As you allude to in your question, except possibly on certain joyous days, your tear your clothes the first time you visit the kosel hamaaravi; some do also when they first see the desolate cities of Judea; some only do this when seeing the actual ground of Har HaBayit.
- If you are leading the prayers (in most of Israel), expect birkas kohanim during chazaras hashatz of every shacharis and musaf. Say each word before the kohanim. However (in most synagogues), do not say "kohanim": someone else will do that for you. If you are not leading prayers, but you are a kohen, remember to go and wash after kedusha.
- As you allude to in your question, t'ruma, et al., are more relevant in Israel. Buy stuff — including fresh produce — only with a reliable hechsher (or from a store with a non-expired reliable hechsher), or CYLOR about what you can buy and how to separate t'ruma, et al., yourself.
- Although in the States people do not use hot water from the tap on Shabas, in Israel many people do. (This is due to a difference in how the heaters work.)
- OTOH, some don't use electricity on Shabas and yom tov at all.
- If you rent an apartment to live in for the end of your trip, realize that the 30 day exemption from mezuzah for renters only applies in the Diaspora. In Israel you must get a mezuzah immediately.
- (Due to reasons of calendar, this is not relevant to the specific case asked about in the question, but will be relevant to others.) If you're to be in Israel for Pesach, there's a dispute about when your chametz in the States or Australia may not be owned by you. Consult your rabbi: you may need to sell your chametz earlier or buy it back later than everyone else.
- In maariv Nussach Ashkenaz omits the third bracha after Keriat Shema (that begins Baruch Hashem LeOlam) which they do say in Chu"l. Additionally, Nussach Ashkenaz adds Morid HaTal in the summertime.
- Some are very careful about not crossing against the light on Shabbos, because they may cause a non-shomer-shabbat Jew who is driving to brake.
- If you're visiting Yerushalayim (and some other places) there will be a Minhag to have candlelighting 40 minutes before Shekiah.
- Be sure to keep in mind that there may be issues with Ribbis or Ona'ah when borrowing / repaying with different currencies.
- If you are there during or immediately following a shmitah year, you will have to make sure to follow those laws.
- There can be differences in the Torah reading cycle if you are visiting after a Yom Tov.
For details and guidance about all of these, consult your rabbi.
I've made this a "community wiki" post so others can add to it (and so that, I think, I get no reputation points from votes on it); please contribute.