How to use switch with more than one variable?

there is no need for multiple switch statement, since you have a fixed file name with each weather condition number in the filename you could just do this

var src = "";
// concatenate todayCondition with the rest of the file name
src = "condition" + todayCondition + "Img.png";

document.getElementById('todayWeatherIcon').src = src;

Note: You should only do this if you know the names of the file won't change in the nearest future

You could just set the condition like so

src = 'condition'+todayCondition+'Img.png'; document.getElementById('todayWeatherIcon').src = src;

You should simply use a function :

function getIcon(weatherCondition) 
{
    var src = ""; //This variable will contain an icon that represents the weather condition.
    switch(weatherCondition){ //weatherCondition is the weather condition it's in the range [0-47]
        case "0":
            src = 'storm.svg';
            break;
        ........
        ........
        case  "47":
            src = 'rain.svg';
            break;
    }
    return src;
}

var day1Condition = getIcon(todayCondition);
var day2Condition = getIcon(tomorrowCondition);
...
document.getElementById('todayWeatherIcon').src = day1Condition;
document.getElementById('tomorrowWeatherIcon').src = day2Condition;
...

If the image names are all different then you can best use an array of strings, like this:

var images = ["cloudy.svg", "sunny.svg", "rainy.svg"];

// Arrays are designed to work with numeric index values:
console.log(images[0]);
console.log(images[1]);
console.log(images[2]);
console.log("--------------")

// Javascript also accepts "numeric strings" as array index values:
console.log(images["0"]);
console.log(images["2"]);
console.log("--------------")

// Or using a variable, this is the closest to what you need to do:
var todayCondition = "1";
var src = images[todayCondition];
console.log(src);