Tableless web design gives you the following:
Less Markup
Tables need a lot of code to actually create table cells and rows, where as a div is only a simple open and close tag. This means less for your users to download when they're loading the page.
Easier Maintenance
Since there's less markup with tableless design, the code is easier to maintain (less code to wade through). It's also easier to add elements to a tableless design because you don't have to go through a table and determine where all the rowspans and colspans need to be added to your existing rows and columns.
Content and Presentation are Separate
This one is key. With tableless design, it's much easier to swap the CSS and completely change the look and feel of a site without touching the markup.
Tables are for Data
Tables should be used when you're displaying tabular data. Their structure makes them very good at defining relationships between headings and the data. As a result they're great for accessibility when it comes to tabular data.
The Catch
Although there isn't a lot to learn in CSS, it'll take a good while to get used to all the browser rendering inconsistencies (I'm looking at you IE...). If you want to get started, I highly recommend you focus on learning about how floats and position behave. Once you have that sorted out, it becomes much easier.