Hi Sally!
As you will see once you get to Bali, the question is more like, how can you NOT find transportation in Bali? You will be assaulted on the street again and again with the question "transport? where you go? transport?" We have parked our rental car on the street and as we were getting out, been asked if we wanted transport!
They will take you places in their cars and on the back of their motorbikes, the latter of which I will do from time to time, but I have never rented a car off the street, but not for the same reason I wouldn't do it in Mexico. These guys are all very safe and I've never heard one single robbery or sequestering story. But I HAVE heard that many of the drivers will drive you to their relatives' shops or restaurants and stop understanding English when you say you don't want to go there.
The going rate for drivers as of last year was somewhere between 200,000 - 350,000 rupiah for a full day, or around 25 to 40 USD. As a Mexican resident, you know to always bargain and never accept First Price!
We have always driven our own rental car on Bali and never had much problem, we pay 80,000 rupiah a day for our vehicle, which is about 9 USD. Again, like in Mexico, you have to watch em when they gas you up at the stations.
Regarding Kuta, unless you enjoy hanging out on the Front Street there in Cozumel, you can probably pick a more desirable location to spend time in Bali. Kuta is where all the Hard Rock Cafe type places are and drunken tourist youth rage. If you're looking for beach maybe you would like the Sanur or Seminyak areas. It's always sunnier and about 20 degrees hotter down in those places than in Ubud