The city of Kota Kinabalu is really not that big, and once you're in the main city area, everything is relatively within walking distance.  From the northern end of town (where Jesselton Point lies) to the southern end (near Wawasan Plaza / Imago) is no more than 3 kilometers.

Taxis in KK have meters, but they refuse to use them.  Destinations are often fixed prices, with no known formula for how they come up the rates. Bargain before you get in. According to the rates set by the Land Transport Commission (localled called LPKP), the RM10 flagfall buys you 9 mins or 3km. After that it's 12c per 18 seconds or 100m. You can use that as a guide to try and negotiate the cheapest rate.

Apps like Uber, and the local version Grab, also work here. In 2016 taxi drivers have been unhappy about it, because they felt it took business away from them. There were a few incidences of unrest and even some arrests of Uber drivers, but ultimately the Malaysian parliament passed legislation that paved the way to legalisation for these apps. It makes taking taxis a lot more pleasant, so try it out.

You can also organise a taxi to take you to places like the Mt Kinabalu National Park.  The taxis that are willing to do this usually gather around the town field, Padang Merdeka, towards the back of the city.  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 - if you have the time - would be the ever popular minibus.

Old buses are slowly being replaced with new ones, so getting one with aircon is now a likely occurence (although still not a given). While an actual bus terminal is awaiting completion near the south of town (many years already), the bus stops are still scattered around town. 

Purple buses go out towards 1Borneo and Sepangar, blue buses go to Likas (stadium and close to mosque) and Inanam (also the Inanam Long Distance Bus terminal for buses to Sandakan). Both these colour buses can be found towards the back of the city near the hill (behind Asia City Complex)

There's also red buses that will go out past the airport towards Lok Kawi, Kinarut, Papar, etc. and orange buses that go to Damai, Lintas, Beverly Hills and along Jalan Penampang past the Museum, Queen Elizabeth Hospital all the way to the KDCA and Donggongon.

There's a specific Airport Bus that arrives / departs from Centre Point Sabah and Padang Merdeka. October 2016 it costs R5 one way. 

Yes, Kota Kinabalu has a train, in fact, one of the oldest lines in South East Asia. The North Borneo Railway operates a scenic run with a steam engine and original, renovated, 1800's coaches, currently on Wednesdays and Saturdays (enquire at your hotel), but also operates a scheduled (more or less), public service that runs between the train station in Kota Kinabalu (near state mosque) and Beaufort, and a section to Tenom (where condition allow). The train journey is interesting, cutting along the coast and into mangrove swamps, and along some padi fields with water buffaloes. Destinations include Papar, interesting for the market and some other things, and onwards to Beaufort.

Long Distance Buses
Bigger, air conditioned coaches, from in front of the Sabah Court House will take you to places like Kuala Pengu, Menombok, Beaufort and even into Brunei and to Sarawak as far as Kuching. From the Inanam Long Distance Bus Stations you can catch a bus to the east coast, to places like Tawau, Lahat Datu and Sandakan. 

Significantly cheaper than flying, it does take a little longer, but the drive is scenic if not somtimes a litte scary, thanks to the periodic over stimulated bus driver. Enquire at the little wooden kiosks about price and departure times.