If you don't mind looking a bit like other sites, I'd highly recommend using bootstrap as a basis for a modern-looking redesign. You can customize it to your will, or even buy some bootstrap-based layouts to work with for not much money: https://wrapbootstrap.com/

As someone who learned table-based layouts and who now designs using HTML5 and CSS3, rest assured that the "new" web is not that hard to learn as long as you understand the basics of css. And almost all your JavaScript skills will transfer. It's really as easy as picking up a book on HTML5 and reading it. Alternately, check out: http://diveintohtml5.info/