There are many ways to collect, store and mount booklets and panes.
To begin with, look at the Scott Specialized catalog. There is an entire section which deals solely with the booklets and convertible booklet panes.
BC listings are for the convertible booklets. Basically, these are the panes of twenty stamps with a strip which can be removed, allowing the booklets to be folded. The other type is the two sided booklet panes, such as the flag issue currently available. Here are examples of both types:
First up is BC145 (the Scott number for the booklet pane). This is the 33 cent Love stamp. The strip can be removed and the booklet folded in half for easier storage. It is best to keep these flat as that is how collectors want them. You'll also note that the plate number is on the removable strip.
The next item is the Hot Rods booklet pane. This is the twenty stamp type with eight stamps on one side and twelve on the other. These have two strips which can be removed and the booklet folded. I keep two of these so I can show both sides of the booklet.
The second major type are the stapled/glued in booklets and panes. These have a prefix of BK in the Scott listings. The older booklets were stapled and first appeared in the last years of the 19th century. These are bulkier and harder to mount as a whole on an album page. One method used to to "explode" the booklet, showing the covers, interleaving and panes. An example of this is shown below, using the C25 Transport issue booklet pane (BKC2 - The addition of the "C" denotes airmail issues).
The last BK listing is for the Harry Potter booklet issued in 2013.
Within this BK area, in the 1980's through the 1990's, collectors could purchase from the Philatelic Center of the USPS what are called "never folded" panes. An example of this is shown next. It is Scott 2474a and would have come from BK171.
Note the plate number is the selvage. This is the key plate number for this booklet pane. The current Durland has it listed incorrectly at $400; it should be $50.00 and will be reflected in the next edition of the catalog.
An area I find interesting are the makeshift booklets which use commemorative stamps which were not produced as booklets. Here are two examples of the many which were created. As eyeonwall points out a little later in the thread, they are also called MDI booklets after the company the USPS hired to create them. (Thanks for the correction, eyeonwall).
First is a booklet created from the Georgia Okeefe stamp, Scott 3069.
Next is the booklet created from the Classic Movie Monsters stamps, Scott 3168-3172.
As to storage, it depends upon which type it is.
For the BC panes, I keep them in either a stockbook for the twenty stamp double sided panes or in a binder in a page protector for the ones like the 33 cent Love pane shown above. Eventually, I'll design the pages to mount these in an album.
The only way to effectively maintain the BK booklets is to either explode them and mount them as shown above or to keep them individually in a glassine envelope in a standard storage box, which is what I do.
Finally, the never folded panes are kept in a stockbook until I design the pages for an album and mount them.
Hope this gives you some ideas.