Where to live in Al Ain

Luca,
I have been in Al Ain for almost 3 years.  You will not find furnished accommodations in Al Ain like you would in Abu Dhabi and Dubai.  If this is important to you, you can try a short term or long term stay at the 3 Al Massa Hotel/Apartments in town.  Two are in the town center/central district, and the third is near Al Jimi Mall.  Search on Google Maps.
Do you know where you will be working?
Living in town will provide you the convenience of walking distance to the shops and transportation.
Electricity and water is same bill...it all depends on the size of the accommodations.  A 2 bedroom flat could possibly be 300 AED in the winter, and 700+ AED  in summer. We all use differently. 
Rent....use web sites Dubizzle to search pricing. I also suggest a google search of Abu Dhabi Commercial Properties(ADCP) as well...they have a property search feature that can be helpful.  Also, get on Facebook...lots of groups under a search of Al Ain.  Remember, they do rent differently here than anywhere I have been...they like rent paid upfront, or in 3-4 payments.  You may get lucky and find a landlord that allows monthly.
Good luck....stay positive. Its an adventure.

David