Probably the best way is to drive around the areas you are interested in and look for signs out the front 'di kontract', also ask in local kiosks and shops. Some areas that have western style housing for rent would be the Batu Bolong BTN area, Senggigi, and Karandangang. I am not up on current prices. 5 years ago rent ranged from 10 Juta per year to 30 juta per year for a basic 3 bed house with no hot water, no air cond and of course, not pool. A local style house was cheaper, and still would be. Given the way land prices and cost of living has increased over that time, I would expect perhaps double or more of those prices.
also, in Lombok, you mostly pay a year's rent in advance. Before entering into any agreement for rent, make sure you draw up a contract with the owners setting out things like who pays maintenance, who is responsible for what. Don't assume it will be the same as it is in your home country.