We have one large fishing tackle box that has several pull out compartments.  It's sort of like this one:

http://www.amazon.com/Flambeau-Outdoors-Front-KwikDraw-Tackle/dp/B000EDRRZO

This is good for taking to tournaments also, to be sure you have extra parts and pieces.  But, to be honest, on a day to day basis, it is cumbersome to drag out the individual containers and look through each one to find the part you need.  So, in reality, most of the parts wind up getting tossed into one of these tall plastic drawer units and the kids just dig through and find what they need faster.  Like this:

That is what most of our Legos are in.  I also have one of these that can mount on the wall:

http://www.amazon.com/Akro-Mils-10124-Plastic-Hardware-6-5-Inch/dp/B003TV3NL0/ref=sr_1_1?ie=UTF8&qid=1408465078&sr=8-1&keywords=plastic+drawer+organizer+tall

That is really awesome because you can access the individual compartments for small pieces faster than pulling out the containers from the tackle boxes.  If you were really organized and handy with a label maker you could even label the drawers for super fast finding and cleaning up.  (I am not!)

We are lucky to have a dedicated space to practice and we don't have to move our stuff, though.  I've been coaching for many years and we have acquired quite a few Legos, so my storage solutions have changed over the years.  I would say the most important thing to think about is can the kids clean up on their own quickly at the end of EVERY practice.  Don't let it pile up or it will require a day long sorting session.  (Though we've had many, many of those too!)  

Sonya