To simplify things, you just divide HK into 4 parts: HK Island,
Kowloon, New Territories, and Lantau Island.
The MTR (Mass Transit Railway) connects HK Island, Kowloon and
Lantau Island.
The KCR (Kowloon-Canton Railway) connects Kowloon, New Territories
(e.g. Shatin, Tai Po, Lo Wu, Yuen Long, etc) and the HK/Shenzhen
checkpoint.
The Star Ferry connects HK Island and Kowloon.
The tram only runs on HK Island.
Taxis and buses are everywhere.
Most of the sights are in HK Island and Kowloon, so most likely you
will only need to take tram, Star Ferry and MTR. MTR is the
fastest, but it's expensive (unless you buy a MTR tourist pass).
Avoid MTR during peak hours.
The octopus card can be used on all mode of transport (except
taxis). When you buy a normal adult card, it costs $150 ($50
deposit + $100 value). To be frank, the $100 value will not last
long. (The trip from the airport to the city already eats a large
chunk.) If you travel solely on tram and star ferry, the card
should be able to last for one week. If you take MTR & KCR
often, you may have to top up $100 every 2 days. If you make a trip
to the big buddha in Lantau Island, you may have to top up your
card before and after the trip!