When I first moved to New York, I remember getting to my apartment which (luckily) had a mattress, but nothing else. I needed to buy things, and quickly so that I could start my adventure from the comforts of a bedroom that was actually comfortable to be in.
What I didn't immediately think of though was that all the stores I had grown up with have different names here (I know, in hindsight, of course they do). So here's my guide (and it will keep growing) to all the businesses you'll need day-to-day.
Everything
Ok, first off, there's Amazon. Sure, it's now launched in Australia and overseas but the U.S. version is the original and the best (and it's HUGE). This will seriously cover 95% of things you need, and if you sign up for Amazon Prime, it's 1 or 2 day delivery for “free” (you pay for a Prime membership, but when you're setting up it's totally worth it).
Also note, I linked above to SMILE.Amazon.com and not just Amazon.com. Lots of people don't know, but Amazon has a charity program where a percentage of every purchase you make on Amazon will be donated to a charity of your choice. Prices don't go up, it's straight up just a free donation. You should ALWAYS use Smile.
Homewares
Bunnings is (The) Home Depot or Lowe's
Target is still Target
IKEA is IKEA
K-Mart doesn't exist
WalMart is here
Office Supplies
Officeworks is Staples or OfficeMax / Office Depot
Entertainment & Electronics
JB-Hifi is Best Buy
Food
In NYC your supermarkets have been replaced with Associated, Morton Williams, Finefare, and C-Town.
Subway is still Subway, but Subways are now also train stations.
There are so many more, comment yours below!
There are 2 KMarts in Manhattan:
Penn Station
Astor Place