A work visa is preferable, but should be applied for at your local embassy. Some schools will help you obtain one. However, it also possible to find temporary work with a tourist visa. The wages are on average 4 to 5 dollars, which is low for western standards, but is a high salary in Ecuador. Because Ecuador is relatively cheap, you can live on this comfortably.
Accommodation is cheap and easy to find.
Many internet cafes have a notice board where rooms or apartments are advertised or otherwise try the Sunday edition of El Comercio. A big room should be well under 100 dollars.