The closest activity to the airport is probably the cable car from the end of the tung Chung line (about 5- 10 mins by taxi from airport) which takes you over to Lantau Island and the big Buddha there. To go into Hong Kong itself the quickest way would be to take the Hong Kong airport express which goes to Hong Kong station and connects to the central and island lines. You could walk around central and see the old colonial buildings there and zoological gardens or take the star ferry over to Kowloon. From central you can take the tram to the peak, but you may spend quite a lot of time queueing.
Yes avoid the peak tram queue, but the peak itself is lovely to visit. You can get a cab or bus up there. Then walk along one of the lovely trails to see how much nature there is, even on HK island itself.
Definitely get the ferry across HK harbour to Kowloon.
Walk around Sheung Wan to see all the amazing food shops and local life.
The public transport is really good so you should be able to get around pretty quickly.