Banking Switch your home banking over to an international bank at least a few months before you leave. HSBC and Citibank are good options. Make sure you have some money successfully wired over before you arrive so your first trip to the bank is as smooth as possible.
To add an extra note for folks heading to Singapore: I haven't seen any Singaporean health plans that don't absolutely suck. Unless your new job comes with a full expat policy from your home country, assume the local policy they are going to offer you is going to be 80% useless. The crazy thing is, your company's HR department will think their plan is great. They also think it's normal for families to spend all of their retirement savings if someone gets hurt. It's a cultural difference and one that unfortunately can get you in some very serious financial trouble.
A warning to women: expat policies usually do not include maternity coverage. Maternity insurance is an additional policy with a 10 to 12 month waiting period before you can claim. Even the most expensive expat policies do not cover women who are already pregnant. Do not leave your home country pregnant or intend to get pregnant in the first half year of your relocation. I've met far too many trailing spouses in Singapore who ended up playing Russian roulette with their pregnancy. Just to throw a few numbers out there, a C-section could put you out $22,000 sg and a premie baby could be around $250,000 sg.
Address change. Moving away when you don't have a forwarding address is really complicated. In Canada you can do int'l mail forwarding but in the US you can't (I guess the US can't imagine anyone leaving?). I recommend getting a mail box at a private mail forwarding company. Set up the address change before you go. Once you arrive, have them bundle mail once a month and send it over. We used Bongo this time and they were awful so just avoid that one.