This display case looks like it can hold about fourteen coins. When you buy your fifteenth, will you buy another display case like this one? These will end up consuming a lot of space.
Tarnish is another word for
patina. However, tarnish is generally viewed as undesirable, whereas
patina is generally viewed as desirable, particularly with
ancient coins. Of course, there can be both attractive and ugly patinas.
Patina is also called
toning.
If your coins are white as white and you want to keep them that way, then they should probably go into
flips. For best results you will want to use acetate "hard"
flips rather than vinyl "soft"
flips. Long-term storage in vinyl
flips can result in the coins developing a gooey green
PVC film which will cause damage to the surface of the coins.
However, even if you use hard
flips, there is no guarantee that the coins won't develop a
patina (or tarnish as you say).
If you are collecting ancients, it may be best to accept that they will develop a
patina over time. Also if you accept
toned coins, it will greatly broaden the
field for you in terms of what coins are available to add to your
collection.