The cheapest and easiest way from Suvarnabhumi Airport to Bangkok CBD

There is a train ever 15-20 mins

Announcements are in Thai and English

Take the airport link train (in Basement) after clearing Customs. The lift is easier if you have lots of luggage but there are escalators also.
You can buy your token from a ticket window. Makkasan Station (connects with Phetchaburi MRT - underground) costs 35 Thai Baht (about US$1). The journey takes 25 minutes. Phaya Thai station is the end of the line (45 baht), this links with BTS Skytrain.

Beat the traffic. View from pedestrian walkway, Phetchaburi MRT top left

MRT map (southbound)

Follow the covered walkway (about 170 metres) to Phetchaburi MRT. This line connects Sukhumvit and Silom Road hotels. Fares start at 16 baht. Ticket machines give change.

The cheapest way to get from Sydney Airport to the CBD

If you are arriving at Domestic Terminal 2 or 3, take the 400 bus (outside Terminal 3) to Mascot Station (the next but cheaper underground station) $2.30 Adult fare. It's the first stop after the bus turns right off Bourke Street. You can buy your ticket from the driver but have the correct money. Or PrePay, buy tickets at shops displaying PrePay signs.
The 400 bus stops at International Terminal (T1) before T3.
At Mascot Station the fare is $3.80 (as of February 2014) to St James (Hyde Park) or Circular Quay (Sydney Harbour).
If you take the train from Domestic underground station (under the terminal) it will cost $16.40 Adult one way.
International Terminal (Airport Link) costs $17.20 for same destination (but if you are jetlagged or you are travelling with 2+ people), I'd taxi it or take that train from International underground station (if travelling solo).
In summary: $15.90 train or $6.10 bus+train.
The same 400 bus continues onto Bondi Junction from Terminal 3. Bargain.