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);
    }
}

@Claude, you are right, the other solution I proposed was too different from original code. This is another possible solution, using setInterval and switching functions:

function onloadFunctions() {
    var count = 0;
    var refId = null;
    var target = document.getElementById("aux");

    var countUp = function() {
        target.innerHTML = count;
        count ++;
        if(count >= 9) {
            window.clearInterval(refId);
            refId = window.setInterval(countDown, 500);
        }
    }

    var countDown = function() {
        target.innerHTML = count;
        count --;
        if(count <= 0) {
            window.clearInterval(refId);
            refId = window.setInterval(countUp, 500);
        }
    }
    refId = window.setInterval(countUp, 500);
}

clearInterval(this);. You can't do that. You need to save the return value from setInterval.

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

var count = 0;
function countUp()
{
    document.getElementById("here").innerHTML = count;
    count++;

    if(count == 10)
    {
        clearInterval(interval);
        countDown();
        interval = setInterval(countDown, 200);
    }
}
function countDown()
{
    document.getElementById("here").innerHTML = count;
    count--;

    if(count == 0)
    {
        clearInterval(interval);
        countUp();
        interval = setInterval(countUp, 200);
    }       
}