Laptops:
Avoid Intel Video Chipsets - Intel Video chipsets often don't perform well playing games. I know it supports hundreds of games, but not the ones you may find on the Xbox or PS3. Many games actually don't mention them on their requirements.
$350 isn't really alot go on. Using a randomly picked Laptop for slightly higher price (UK £300) doesn't have the correct Graphic card for games released last year.
PC:
$350 isn't really alot go on, but you can buy in parts and get better PC in time. Some Components (for example: Graphics Card) can actually exceed $350 by themselves. However, you can buy the components and then build your own Pc, you can upgrade over time. Many off the shelf PCs, such as Dell built ones, do not make it easy to upgrade.
If you have a Desktop PC you're getting rid of, see what parts you can recover, this can make the initial $350 Purchase go further. I've been doing it 10 years, but there where times where you really need to spend lots of cash to get it going.
If you want to play recent games, one of the most important piece of hardware you'll need is a Graphics card. A lot of games now use the Graphics card more than the Main Processor. Don't skimp out on the Main Processor, games still need to use it. Some Motherboards come with integrated Graphics cards, I'd personally avoid them. They are often low powered, and end up being useless in the long run. nVidia and ATi/AMD make decent graphics cards.
Unless you're planning to use Linux or you have a copy of Windows Vista or Later to use, you will need to buy copy of Windows, if you're building your own PC. Some modern games today want a 64-bit version of the operating system, and you will need it if you have more than 3GB of memory.