Step 1: 2"X2" s
One of the most popular types of coin storage is a 2"X2" cardboard container lined with clear mylar. The coin is placed on the mylar, and the case is folded and stapled shut. With this method, the coin will not slide around in its case while handled.
Pros: Inexpensive, easy to label, compact.
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Don't be fooled that slabs are airtight and therefore the best way to preserve a coin - they are not. Although sonically sealed, coins in them can, and do tone. Humidity can, and will alter a coin in a slab. Slabbed coins also need to be stored in airtight containers to hinder toning.
According to the (many) dealers I have talked with, the 2X2s holding the coins will preserve them best (not including Air Tites and similar non-name brand holders like Air Tites). A Mylar flip like on your silver Eagle is next best.
The Mylar flip has an air pocket surrounding the coin, and therefore allows more airborne humidity/particles that might harm the coin to
contact the metals surface area. Whereas a 2X2 presses the coin between two sheets minimizing exposed surface area of the coin.
If they are not to be displayed, then put the coins (in their holders) in an airtight container with a silica gel pack (ask a local, private furniture store as they get packs with kits they have shipped to them and only throw them away). Zip-lock bags are not airtight over time.
When displaying coins, it is likely best to put them in a 2X2 (or the Mylar plastic flip like on your Silver Eagle), and then get some of the pages made to hold 2X2s that fit in a 3-ring binder. Even the more expensive coin albums hold coins in cardboard-sided holes with plastic covers. They look nice, but again, the cardboard will tone the coins as it absorbs/retains humidity.
BTW - PVC plastics will harm silver coins over time (green slime forms over time!). Make sure when buying holders that they are PVC free. PVC is not normally used in modern coin holders, but some old albums did use.