How do I correctly use setInterval and clearInterval to switch between two different functions?

You need to capture the return value from setInterval( ... ) into a variable as that is the reference to the timer:

var interval;
var count = 0;

function onloadFunctions()
{
    countUp();
    interval = setInterval(countUp, 200);
}

/* ... code ... */

function countUp()
{
    document.getElementById("here").innerHTML = count;
    count++;

    if(count === 10)
    {
        clearInterval(interval);
        countUp();
        interval = setInterval(countUp, 200);
    }
}