A place in Quito

For permanent stay, Olx is a great way to get started to get an idea of prices for different areas.

For a longer but not permanent stay, ideally renting a room would be the best option, but I'm not sure about dogs being allowed. AirBnB is a great and convenient website to rent furnished temporary apartments. We used them a few times here and elsewhere and it's practically hassle free, and money is never exchanged as you pay direct to AirBnB and they deal with the landlords.

As for which areas, that is completely relative as Quito has such diversified neighborhoods. There's the historic area which I don't like but others love. There are quiet enclaves like Tenis which is very modern but too quiet. There are nice middle class neighborhoods like Rumipamba and La Granga which are not as pricey yet close to many amenities. There is also the valley which has about about an 8-10 degree temperature (higher) difference and lesser altitude yet only 20-30 minutes from Quito.

Then there are the much northern areas which is cheaper, but distant from the center of the city and can be as much as an hour bus ride with traffic. As for the south it is an arbitrarily built, densely populated concrete jungle with a much higher crime level than the rest of the city.

So there's north, central area, south, and the valley areas.