Please recommend 3-4 search websites for cheap flights. Also is it true that as you get closer to the departure date, prices will decrease because airlines are desperate to dump seats?
I do most of my searching on kayak.com This year I bought my tickets about four months out and they dropped about $100 five weeks before the trip. I wouldn't wait much later than that to buy.
If there are seats to dump, then maybe. There is a lot of folklore around when to get cheap seats and most of it is worthless. The risk that you run by waiting is two fold tickets not there or more expensive. Second, waiting to the last minute may result in less desirable routing and seating. Third, you may find it cheaper to into and out of major international airports like Milan and Rome. Sometimes the extra leg to Venice and Naples can add substantially to your ticket cost. Set your budget and buy when it is near that. There are no magic ways to find the cheapest flight.
I like to buy 3-5 months out. I feel like I get the best mixture of affordability and selection.