Not much of a home remedy, but if I ever feel even a hint of sore throat/coughing or whatever I immediately start ODing on vitamin C (not literally. Also literally doing so is very hard anyway. Also also not a doctor, etc.), start chugging water, I try to down enough to keep my stomach full and keep over drinking whenever I take a normal drink, if anything hurts I take DayQuil or if unavailable I take acetaminophen or ibuprofen in that order. I also experimented with gobbling zinc (normal consumption amounts in throat lozenges).

When I had a "you can take zero days off!" job I went 4 years and I don't think I ever took a sick day. One time my boss literally forced me to go home. (Taking sick days is good/normal, I was just being a young dipshit, but the vitC, water, pain management combo definitely helped me through 99% of common cold symptoms)