When I was searching for apartments last year, I spent a lot of time on the internet looking for places. Then when I went out driving around to actually go see them, I noticed apartment buildings all over the place that I had never seen online.
I would definitely suggest just going out for a drive in the area you want to live and looking for places that way. Unfortunately, I found that this technique doesn't work nearly as well when you're trying to rent a house.
Depends on the market. Some places in the US, realtor.com will have fairly comprehensive listings. In others, it's a dud. Real estate firms which specialize in property management can also provide a lot of options on their sites.
The issue is that the data are typically fragmented and often proprietary and aggregation websites are focused on a revenue stream from the listing agent rather than the renter.
Padmapper was a good solution that integrated everything but then Craigslist sued them, not sure on that part anymore.
Craigslist is still ultimately king but if you want to be comprehensive, you might need to check a few different places. Local newspaper websites are good too.
apartmentguide.com and apartmentratings.com have been all I've ever neede. But if you want to disrupt, knock yourself out.
If you're in Los Angeles everyone I know is using RadPad (http://onradpad.com). It's an iPhone app that is like a cross between Instagram and Craigslist. It makes looking for a place so much easier.
Craigslist, phone calls, a camera, and enough cash to secure a lease. Depending on how picky you are, finding the listings isn't the bottleneck, it's going and visiting 30 different places.
Sometimes walking around outside in the area you want to live in works. Sometimes asking friends on or off of Facebook works.