To be truthful, I don't think you were getting the best results from your beer with you previous bottling method. After you aerate your wort and pitch the yeast, you want to minimize oxygen exposure to the wort. Bottling beer typically introduce oxygen, but you want to try and minimize it as much as possible. Putting your finished beer through a funned or screen will increase the surface area or your beer and introduce a lot of oxygen. Oxygenated beer will lose hop aroma very quickly. It may not be present at all when the bottles have reached full carbonation. Also, the beer could go stale very quickly and introducing other unpleasant flavors such as cardboard.
Transferring to a keg is not too different from transferring to a bottling bucket. First you need to clean and sanitize your keg, auto siphon, and transfer tube. Then put your fermenter on a counter top or table and leave the keg on the ground in front of it. Then start your siphon from the fermenter to the keg. You want the transfer tube on the bottom of the keg so that it fills from the bottom up and try to eliminate splashing as much as possible. Hold the auto siphon so the bottom tip is well above the trub at the bottome of the fermenter. Keep it suspended in the clear bear and lower it as the fermenter drains. Once you get to the trub at the bottom, stop transferring the beer. You can also tilt the fermenter to one side a bit to try and get the last bit of beer, but that takes some practice. Transferring this way should eliminate the need for filtering and your beer will have less oxygen exposure.
There are other methods to transfer to a keg, but based on your post, I'm guessing that you have the equipment to do the method I described. A cheap and easy upgrade might be to add a spigot to your fermenter. This was my original method before changing to a closed transfer. With a spigot, you simply attach the transfer tube to the spigot valve and open the spigot, eliminating the need for an auto siphon. After a couple of batches, start reading up on closed transfers. There are different methdos to do this, but the simplest is a closed gravity transfer which doesn't require additional hardware, just a spigot.
