The easiest way of getting around Kota Kinabalu is by taxi.  Once you have reached the city centre, everything is really within walking distance.

From the one end of the city centre (the harbour) to the other end (Wawasan Plaza) is less than 3kms, and if that's the city's lenght, it's less than 1km wide.  Built roughly along a grid, you can hardly get lost.  The only challenge is braving the sun and the heat if you choose to walk.  However, taxis are sufficiently plentyful and cheap to avoid that labourious task.

For the outer reaches of the city, taxis are still the preferred method, although you can also take a minibus from the terminal next to Wawasan Plaza.  Rides are dirt cheap, but is crowded and not air-conditioned.  For the decirning traveller, the crowds and pace of the bus might be a little alarming.

For excursion further afield, an organised tour is best (unless you know what you're doing). This can easily and cheaply be arranged by your hotel conceirge.

A good way to start your stay in the city is with a half-day city tour, if for nothing else, just get orientated and perhaps be exposed to some sites and sounds you might otherwise miss.  Choose the option that includes the dinner & culture show (usually called KK By Night), which is informative, interesting and entertaining. Oh, it's also quite cheap, so why not.