It is best to establish what your budget is, then determine what your options are.
If your not planning on having personal transportation, then you want to find something in or very near centro.
You can find apt from the low end of $250 up to $1,000 usd a month. Average for a decent place is around $500-600 a month. Utilities are extra. Your electric bill will be the highest, and in summer can be higher then your rent.
As to "living here" you can come on a tourist visa for up to 180 days. However, you can not work during your stay. You can apply for a temporary or permanent residency, but you will have to document that you have enough funds to provide for your residency. You will be allowed to work, but your employer would have to document what exactly the job is, then that is submitted with your application.
You can NOT apply for temporary or permanent residency here. You must start the process in the US for a US citizen.
My advice is to try it for 6 months on a tourist visa. If after 6 months it's what you want, go back to the states, and start the process.