You should put the bread in the freezer instead of the fridge if you want to store it and keep it fresh for a longer period of time.
Refrigerating bread can in reality make the bread go stale faster and freezing slows down the process of the bread becoming stale and preserves the quality longer.
For short term storage of bread it's best to store the bread at room temperature in a breadbox or other air tight container for a few days.
Bread is best stored at room temperature away from high humidity and while you can put bread in the fridge the bread can go stale much quicker.
The best place to store bread in the kitchen is in a cool dry area of the kitchen away from any moisture.
You could store the bread in a kitchen cabinet away from moisture or in a deep drawer.
Another option is to store the bread in the microwave when not using it or a bread box.
If eating the bread within a few days you can simply store the bread right on the counter in the open and it will be fine for the few days.
In general, most loaves of bread will last up to a week at room temperature, and 3 to 5 days longer in the fridge—though keep in mind that refrigeration of the bread can make the bread go stale.
Bread that goes stale is still useful for breadcrumbs, croutons, and bread pudding.
Using an airtight container will preserve the moisture in the bread and keep it from drying out too quickly.
However too much moisture can lead to the bread molding.
If part of the bread is moldy you can remove the moldy parts of the bread and eat the rest which is what I do.
A freezer bag is also a decent short-term, room-temperature bread storage solution.
Storing the bread on the countertop in a plastic bag (or well-sealed plastic wrap) will help keep it from going stale, but be warned the crust of the bread will suffer and grow mold due to trapped moisture.