we do have separate lorries with different compartments, paper, cans, food waste is a separate truck as is green waste. I pay a couple of quid for a whole roll of recycling bin-sacks. In the kitchen I don't use the caddy, I have a separate pedal bin which is catproof - the sack goes in there then every few days it goes out to the little green lockable bin outside. We only ever have a couple of sacks per week, as any veg waste goes in the compost bin in the garden. The compost bin takes the toilet roll tubes and eggboxes due, they compost down really well. Teabags go in my food waste - I can bear finding half-disintegrated teabags in my compost!
We have got a good system going. We have two separate waste bins in our living room - recycle and not, saves sorting outside - and four in the kitchen now (recycle, not, food and compost).