The city of Kota Kinabalu is really not that big, and once you're in the main city area, everything is really within walking distance.  From the one end of the town (where the harbour lies) to the other end (near Wawasan Plaza and the mini-bus station) is no more than 3 kilometers.

As such, the most popular form of transport are taxis.  They do have meters, but often trips are negotiated upfront.  Destinations like the resort hotels are often fixed at either RM10 to the Sutera Harbour Resort and RM15 to the STAR.  Any other trip will be around RM5 or RM6. After midnight and before 6am, it's the meter plus 50%.

You can also organise a taxi to take you to places like the Mt Kinabalu National Park.  The guy willing and waiting to do this usually gather around the Millimewa supermarket in the centre of town, near the Sugar Bun Plaza.  They usually ask a flat fee and unless you want to pay the whole cab, will wait for the car to fill up with four people. 

Mini Busses
If you want to go to the outlying areas like Tawau, Inanam, Penampang, Luyang, Beverly Hills, etc. a cheaper option would be the ever popular minibus.  Not the most comfortable form of transport, especially as it isn't air conditioned, but it's cheap. Depending on your distance it's around RM4 or less.  The mini-bus station is at the end of the waterfront near the Wawasan Plaza and in close proximity to the bridge they're working on.  Find your bus, hop on and wait to go.

Yes, Kota Kinabalu has a train, in fact, one of the oldest lines in the region. The North Borneo Railway operates a scenic run with a steam engine and original, renovated, 1800's coaches, every so often (enquire for details at your hotel), but also operates a scheduled (more or less), public service that runs between the train station in Kota Kinabalu and Beaufort. The train journey is interesting, cutting along the coast and into mangrove swamps, and along some padi fields with water buffalos. Destinations include Papar, interesting for the market and some other things, and onwards to Beaufort.

Long Distance Busses
Bigger, air conditioned coaches, located in the vicinity of the Golden Screen Cinema complex, will take you to the further places like Tawau, Lahat Datu and Sandakan.  Significantly cheaper than flying, it does take a little longer, but the drive is scenic. Enquire at the little wooden kiosks about price and departure times.