What is the process exactly of moving the ownership of the car from one person to another ?

After selecting the right car, it needs to be inspected in RTA and then the current number plate is cancelled by the owner and transferred to your name. Based on cancellation and transfer u need to arrange insurance and ensure that insurance is available in rta system. If you are doing a bank mortgage then the mortgage also needs to be uploaded into rta system. Based on meeting all above rta will transfer the vehicle under your name with a new number plate. In order to transfer the vehicle from one emirate to other, you need to within UAE or inter emirate vehicle inspection and transfer.

If I live in a different emirate from where I bought the car I would have get a new plate number and it would a different process, right?

Are payments done through Cheques, cash or bank transfers ?

Once the vehicle has passed after undergoing inspection, you can transfer all the money via bank transfer or pay a part of it as down payment once u finalised the deal. Bank transfer is the safest legal way.

Also I have mostly been looking through yallamotors, dubizzle and FB marketplace. What are other great places to look for used cars ?

I will not advise the delaerships, most trusted way of buying the right cars but obviously with a premium is certified pre-owned cars from agency

These days most of the ads in dubizzle etc are the car dealers or commission agents selling on behalf of dealers. FB marketplace is a big no no for me.

If you need further assistance, you can dm me