I'm not sure about best times of the year - I only moved here this year! However, while it's best to have a look to make sure you can find things in your budget a few months ahead, don't expect a flat to stay on the market that long. I found mine in the first week of viewings, after a week of proper searching, and we moved in 3 weeks later. I'd say no further ahead than 6 weeks or so.
As for sites, check out zoopla and rightmove, both good sites. The agencies differ depending on what area you're looking in. All I can say on that one is that we moved with Mann Countrywide, and while their fees were a little on the pricey side, they were really good.
Hope this helps, best of luck in moving!