Scroll to today by default

Is there a way that I can set the focus always to Today without pressing the "Today" button? 

Is there a script I can add with the same functionality?

Yes, put the following script into Widget Properties > Advanced > Gantt Config Script:

gantt.attachEvent("onLoadEnd", function(){
    gantt.showDate(new Date());
});

As an alternative, you can also use the following script:

gantt.attachEvent("onDataRender", function(){
    gantt.showDate(new Date());
});