The short answer? It’s up to you. Stamps can be mounted in albums on gridded sheets and neatly labelled, although many choose stockbooks or use Prinz or Hagner pages which are more flexible. Mounting may involve the use of hinges, but increasingly Prinz or Hawid type mounts have become popular as they do not damage the stamp, particularly the Gard version.
The arrangement of collections will largely be determined by your collecting interests and/or specialism. Collections might be arranged by country and in date of issue order. Some bird stamps are part of larger sets that include non-bird issues, perhaps a general wildlife or nature set. Some collectors will mount the complete set, others might want to only use the bird elements. Generally, we would advise against breaking sets up.
Sometimes it is a pick-and-mix arrangement. Mini-sheets tend to be large and are better kept in Prinz or Hagner sheets rather than mounted along with the rest of your collection, but they can be mounted if need be.
Covers can be either kept in separate plastic slips (dealers tend to sell them in this form) or placed in special cover binders, often called First Day Cover or FDC albums. If you become interested in postal history you might want a flexible arrangement so you can study the covers over time.
But there is one don’t. Don’t store your collection in shoes boxes or biscuit tins. Do something with them! They deserve to be accessible and seen. It should be a collection, not an accumulation.