Jon,
Welcome to the Serious Play community! I agree with Joyce – at most LSP facilitator training sessions those questions are dealt with thoroughly.
Anyway – it depends on an actual session. In most cases I do one day of a simple session where we start off with Window Exploration Bags or Starter Kits. If it is a quick warmup then I let people keep the Window Exploration Bags. However, for more serious starter I hand out the Starter Kits, which I request to repack. I thereafter usually continue with whatever training. The warmup exercises are separate from work with larger kits.
Thereafter I frequently bring Landscape and Identity Kit mixed with Connections Kit for a more serious group-work. This large kit is packed in such a manner that most of the smaller items are kept within assorted trays and most bricks, duplos, animals, pillars etc are just kept either in large piles or storage boxes where people can rummage around to identify the most suitable items.
For storage boxes the most adequate solution I have found is this one produced by Lego: http://www.amazon.com/dp/B0085Y3LKE/?tag=seriousplay-20 – they work well. One of those is spacious enough to house bricks for a pretty large group – e.g. you can fit your landscape kit together with connections kit comfortably in 5 of those storage boxes and these boxes can be operated more comfortably than the original cardboard boxes of the Landscape and Identity Kit and Connections Kit.
The storage boxes close with a nice ‘click’, which makes them safer to transport around – they don’t fall apart even when one should fall on its side. They have small holes in the bottom – i.e. you may comfortably wash the bricks after use and leave them for drying.
For trays I mostly use those: http://www.amazon.com/dp/B00006IA5D/?tag=seriousplay-20 – in terms of measurements they fit perfectly on top of the storage box and the slots are comfortable to fit small items.
Hope it helps and good luck with your next steps!
Marko