A word on terminology: Quiteños don't use the Old City/New City designations coined by the English-speaking tourism industry. The colonial heart of the city is the Centro or Centro Histórico. To the north lies an amorphous sprawl of modern neighborhoods, most notably comfortable Bellavista and La Floresta, and the bustling La Mariscal.
Another word on maps: Quito's north-south elongation makes it a difficult fit for maps, most of which rotate the orientation 90-degrees clockwise. North usually lies to the right of the page on city maps.
Quito has two parallel systems of address numbering. The official scheme employs small green-and-white signs affixed to every building using directions denoted N, S, E, and Oe (for oeste, or west) and followed by a number to denote distance from the city center. Locations in the outer reaches of the city always express their addresses this way; in the center of the city, people stubbornly stick with the old sequential numbering system.
To help you navigate all this, the phenomenal Quito Turismo, a joint venture between the city government and its police, operates several tourist offices or stands in the city, and offers guided walking tours of the Old City. The Ministry of Tourism's iTur office sits in an out-of-the-way location and has little but a few maps and brochures. A private membership organization of note is South American Explorers, with an amazing selection of information about Ecuador (as well as branch clubhouses in Lima and Cusco, Peru, and Buenos Aires, Argentina).