Portable or not?The single biggest decision perhaps is whether you need a portable computer. Laptops make up the bulk of computer sales these days because they're a lot more versatile than desktop PCs. However, for productivity, there's still nothing that beats a desktop computer with a large screen and a decent mouse plus keyboard. While a laptop is easy to take with you -- its small screen and keyboard can be a hindrance to getting real work done if you can't hook it up to a bigger screen and keyboard.
Unless you want to play the latest AAA video games at the best quality settings, there aren't really big requirements for more casual online video games. Casino games like those found on Vernons do not require a high-end PC at all. The average processor will be more than enough for this sort of entertainment and you also won't need a high-end video card. In fact, basic integrated graphics will be more than enough.
Get the SSD!The biggest advice I can give is to definitely buy a computer with a solid state disk. Fortunately, a lot of PC already have a SSD by default but there are still some models that use a hard disk drive. Solid state storage will give you a much more pleasant experience, it's definitely one of the best investments you can make if you want a PC that feels fast and snappy.
Other than this, a decent IPS-based screen is a worthwhile investment. You want great viewing angles, good brightness, colors that look vivid, and a decent contrast ratio. Screen size is a personal choice of course, some people are very happy with 24" while others prefer bigger screens.
The biggest lesson perhaps is that for basic, casual gaming, you don't need a whole lot of processing horsepower.