we just use a mesh screen also(window screen size) and just throw it away when it gets clogged and replace it... one guy used to eat the chunks but he quit...
Well, you can get kind of fancy here - there are different types of traps available to catch things like that. One thing you can do fairly easily is make a settling tank that the drain feeds to then use a small pump to put the water down the actual drain.
Something like this, if you get my meaning - basically a square tank under your sink that the water drains into with a divider that's not quite as tall as the tank. Put a small sump or fountain pump in the secondary side & pump the water down your drain or, better yet, through a filter bank & then down the drain.
Yeah, it's definitely good to keep as much solids as possible from getting into the drain. You can pick up a small pump & float switch at home depot. You can even achieve something similar with a couple plastic buckets. Use a hole saw to cut a hole at the same height in both buckets then glue a piece of 2" pvc pipe into the holes to connect the buckets. Feed the drain from your sink into the first bucket, and pick up a small sump pump at the hardware store - one with a built-in switch to start & stop it when water is present. Put a block or something under the pump so it's not on the very bottom of the bucket, that way it's less likely to pick up larger debris. Most of the larger stuff should stay in the first bucket, so just clean it out every so often & put it in the trash.
If you pick up a couple cartridge filters from the hardware store as well you'll have an even better system - feed the pump into the cartridge filters & use something like a 100 micron in the first one, then maybe a 45 in the second, then put the water out of the filters down the drain. The treatment plant will be happier that way.