If I have to throw away bread, can I throw it away in the regular garbage or does it have to be wrapped separately? I have heard that bread possesses a special status in this regard.

ב"ה Shalom Although we have a general prohibition in regard to wasting food, indeed Halacha gives a special status to bread. (See שולחן ערוך או"ח קעא, א). In principle, it is only a piece of bread that is larger than a "c'zait" which requires to be disposed of in a proper fashion, nonetheless the Shulchan and commentaries point out not to step upon crumbs even less than a "c'zait". (שולחן ערוך או"ח ק"פ, ד.) However, when the bread is spoiled it may be disposed of but it should be done in a respectful way such as you mentioned by placing it in a bag. All the best. Chodesh tov