Small dokar (pony carts) are still seen in parts of Denpasar and Kuta, but they're uncommon. Treatment of the horses is a major concern and there is no good reason to go for an expensive tourist ride.
Bemos are normally a minibus or van with a row of low seats down each side and which carry about 12 people in very cramped conditions. They were once the dominant form of public transport in Bali, but widespread motorbike ownership (which is often cheaper than daily bemo use) has caused the system to wither. Expect to find that getting to many places by bemo is both time-consuming and inconvenient. It's uncommon to see visitors on bemos in Bali.
Bemos operate on a standard route for a set (but unwritten) fare. The minimum is about 5000Rp. If you get into an empty bemo, always make it clear that you do not want to charter it.
Most towns have at least one terminal (terminal bis) for all forms of public transport. There are often several terminals in larger towns. Terminals can be confusing, but most bemos and buses have signs, and if you're in doubt, people will usually help you.
To travel from one part of Bali to another, it is often necessary to go via one or more terminals. For example, to get from Sanur to Ubud by bemo, you go to the Kereneng terminal in Denpasar, transfer to the Batubulan terminal and then take a third bemo to Ubud. This is circuitous and time-consuming, which is why so few visitors take bemos in Bali.
Around towns and along roads, you can always get a lift by ojek (a motorcycle or motorbike that takes a paying passenger). Formal ojek are less common now that anyone with a motorbike can be a freelance ojek (stand by the side of the road, look like you need a ride and people will stop and offer). They're OK on quiet country roads, but a risky option in the big towns. Ojek are more common on Lombok.
Fares are negotiable, but about 30,000Rp for 5km is fairly standard.
In heavily developed parts of Bali, Go-Jek is a popular mobile application that allows you to order on-demand motorcycle rides (in addition to just about anything you'd like delivered to you). Note that you must have an Indonesian SIM card, and it may be hard to get picked up or dropped off in heavily touristed areas due to territory rivalries between local drivers.