Heres what my team does:
1. Use a plano box for storing and separating all the bits as suggested earlier in this thread. It works great. A tackle box as mentioned will work too. My wife sells tupperware and they have a variety of nice containers with bins that are useful also.
2. Get rid of as many "fiddly-bits" as possible. I use D10 for money, clues, stamina and sanity. A different color for each item. Each player gets 1 Red D10 for Stamina, 1 Blue D10 for Sanity, 2 green D10 for Clues, and 2 black D10 for Money. This saves an incredile amount of time in setup and play as you dont have to get change, etc... when adjusting each item. They also store easier in separate bins or in one large one.
3. A while ago, someone posted nice card holder tuck boxes on the old forum or even on BBG. I am sure you can still find them on BBG. Basically they are small tuck boxes you print on your home computer and glue together. When storing the game just put each stack in thier own tuck box for protection. They are simple to make out of heavy weight paper.
4. Get a drawstring felt bag for a monster bin. Its a sinch to closed and keep the monsters stored in as well as useful for play. may want to save a bin in one of you plano boxes for monsters that dont belong in the base game so you dont have to separate them later as they are not marked. there is a list somewhere of what monsters come in each expansion. Print that...its extremely helpful.
5. The plastic bin things that come with in the game can be easily modified for use. Simply cut off the small bin section away from the large bin section. This will leave you a long black plastic piece with 4 bins in it. We use this as a "Mythos holder." One of the large bin on the end is for Mythos cards, the bin next to it that is the same size is used for Gate cards. Simply take the decks and place them in the bin on thier sides. This makes it easy to draw from the front, and discard to the back. Active Environments simply get stood up at the short end of the bin between the cards and the bin side facing the board. When a new environment is drawn, we discard the active one to the back of the deck and stick the new one in its place. The 2 smaller bins in this are used for gate tokens and the little plastic stands for monsters. Everything for the mythos phase except clues are in one convienient place that is easy to draw and discard from.
6. Get some of those business card holders from staples. They work great for the Arkham location decks. There are 9 locations for the base game so we got a couple of these holders. They come in various sizes. We got one 8 slot holder (4 slots next to each other) and another 4 slot holder. This is sufficent for most game, but you can buy more or one bigger one. This saves a ton of room when playing. This idea was posted in the old forum so I can not take credit for this one.
7. small zip lock bags are also useful in conguntion with any of the above to further help separate small bits. (sliders, actrivity tokens, etc...)
8. The small see-through plastic boxes that dice come in when you buy dice from a game store work great for some token storage also. These are generally tall and have a black lid on the bottom. We use one for the clue tokens and one for doom tokens.
9. Last one. This is not a storage solution but we find it very useful. Get yourself a rolling bin. A bowl of something to roll your dice into so they dont go flying about the board knocking things over. When we are at my buddies house, we use his wifes waterford crystal wine chiller bin. It makes a lovely sound when rolling into it and is only about 4" round and less than 2" deep. Wonderful. At my place we use a small plastic tupperware bowl. not as nice as the waterford and doesnt have that musical ring when rolling but serves the purpose.