Central Ubud can be covered on foot, but you will need some form of transport to explore the extended vicinity.
Ubud is generally a little quieter and the streets calmer than the more urbanised parts of Bali. So whilst traffic is slower than in downtown Kuta for example, the sidewalks are often blocked by motorbikes, or a collapsed section necessitates a step off the sidewalk potentially placing you in the path of traffic. That traffic could be a tricycle or a truck, so keep your wits about you.
No metered taxis operate in Ubud, and any you see will be dropping off passengers from further afield.
By Local Car
Most local transport comes in the form of SUVs or minivans that can be hired with a driver for specific trips. Look for the circular yellow “E” logo on the windshield certifying them as Ubud Transport Association members. You can (and should) haggle a bit over the price, and pay less than for the equivalent journey in a metered taxi. A short trip should be not more than Rp 30,000, and drivers will be glad to wait for you for a return fare.
Also, there are guys on motorbikes who will also offer bike rides (ojek) and are about half the price of those in a car.
By Hotel Car
Many hotels are located out of town, and are happy to offer regular, complimentary drop-off and pick-up services to central Ubud. Expect to pay higher than taxi prices if you are intending to go further afield.
As elsewhere in Bali, motorbike rental is widely available, and you will not be short of options. Riding a motorbike in Bali is not for the faint-hearted though, and unless you have an international motorcycle license, not strictly legal. This option should probably only be considered if you are an experienced rider in your home country. Expect to pay around Rp. 50,000 (AU $5) per day for a late model motorbike in good condition. Look for rental agencies on all the main streets, or ask your hotel to organise for you.
You can rent bicycles for about Rp 20,000-30,000 per day. There is a large selection available at the corner of the football field on Jl Monkey Forest. The roads in Ubud are not ideal for bike-riding, as there is traffic and no designated bike lane, however riding through the rice-paddies is a lovely way to spend an afternoon. Ubud is quite hilly, so make you are prepared for a bit of a workout!