I live in Australia and was wondering how the Diablo III's regions affect which server I can connect to, and what RMAH I can use. Do I have the choice of South-East Asian servers (I hear pings are lower to them, as they're in Singapore) or are we all on American servers? How is the RMAH limited within a region?
Diablo III's regions affect a number of things in game:
Your selected region affects which server farm you connect to. The Americas region's servers are located in the USA, Europe's are (obviously) in Europe, and Asia's servers are in Korea. Australian and NZ players will be assigned to the The Americas region by default, and hence the USA server farm. There are no South-East Asian (Singapore) servers for Diablo III.
The Real Money Auction House (RMAH) is region locked - the only RMAHs you have access to are those for the region your Country of Residence resides in. However, you may still have access to multiple RMAHs, as RMAHs are also divided by currency. This page lists the RMAHs you can have access to depending upon your Country of Residence. To use Australia as the example, the USD RMAH will have more items because of the US's larger player population, but the AUD RMAH will not have any currency conversion fees (source). These currency-based RMAHs are separate, i.e. an item on the USD RMAH will not appear on any other currency's RMAH, even those in the same region.
The Gold Auction House is not locked within a region, but is region-specific, i.e. the European Gold Auction House cannot be accessed to provide items for characters in the Americas region.
Players and items cannot be transferred across regions. Any item you buy on a RMAH or Gold Auction house cannot be transferred to a character in a different region.
Note that which version of the game you buy does not affect which region you're assigned to at all, i.e. buying a UK version of the game and playing in the USA or Australia will not mandate or even default to the Europe region. The default region selection is based off your Country of Residence in your Battle.Net account's settings.
Australia/NZ etc are limited to the US based RMAH. You can play on Eur/Asian servers (some period of time after launch, once the game has been patched with said capability.) however you are incapable of accessing their RMAH if you decide to play upon them, only the gold based one.
It's possible to purchase an Asian or European specific version of the game to have access to their RMAH, if you are so inclined, however.
The region you belong to is tied to the country in your account (you can see it on the account management page).
Since you said you're in Australia, that should mean your account is tied to the US region. That setting won't limit the regions you can connect to, so if you want you can play on the Asian servers if the latency is better. However you can only access the RMAH on the region you belong to. To access another AH you have to change your region setting (contacting the Blizzard's Customers assistance), or create a new account tied to that region, but that would mean buying another copy of Diablo III.