We regularly use Booking.com (affiliate link) and Airbnb to find our accommodation. If we’re going to stay somewhere for a shorter time (less than a week, for example), we’ll usually book in advance. When we do 2+ month stays during the off-season (Split, Croatia), we tend to test out an apartment for 2 or 3 nights via Booking.com, then negotiate with the owners in person for a long-term rate. We’ve found long-term rates to be up to 80% less during the off-season!
Note: If you’re new to Airbnb and sign up using this link, we both get $25 in travel credit. Some readers have mentioned that Airbnb modifies the promotion from time to time, so the actual figure shown to you at sign-up might be a bit more or less.