(Skip down to the Summary Recommendation for the tl;dr types)
trk@not halpful suz

The reason you're getting so few good answers is there's so many choices at this price point it's hard to know where to start narrowing it down. I find it's most helpful to give people a finite range of options and let them decide based on how much cash they're willing to put out or other things like brand preferences.
________________________________
Caveats:
I can't in good conscience recommend anything at the $600 point for gaming. You're pretty much stuck with integrated graphics at that point, or maybe something negligibly better like a Radeon 6400/Geforce 300 series. As soon as you say the word "gaming", in any capacity, you need to plan on $700.
Be aware any recent drivers from AMD have issues with older OpenGL based games (read: JKA) which will require the yews of 1) old drivers or 2) other workarounds; so you may want to consider nvidia graphics if you want ease of yews in that area.
______________________________
Features/Performance Analysis:
It looks like standard features for this class include the following: 15" @ 1366x768 TN panel LCD, dual channel DDR3 RAM, 5400RPM HD, DVD+-/RW, at least one USB 3.0 port, 802.11n Wifi, 1Gb ethernet, HDMI and VGA out, 6 cell battery, and 1.3-2.0MP integrated webcam.
Going down the benchmark list, I eliminated anything with a 3dmark06 score of less than 5000. My Geforce 7900 GS from 2007 gets around 4000, and it's rather limiting in some games.
The single most important upgrades I would recommend are 1) better GPUs (duh) and 2) 7200RPM HD. The latter may be somewhat surprising, but I couldn't believe how much faster my 4 year old laptop felt after going from a slow Seagate 5400RPM to a Western Digital Caviar Black 7200RPM. It was enough of an upgrade to make me put off buying a new laptop for a year (I don't really game on it any more, just yews it for productivity). This may be the biggest factor in your request for "a reliable computer that can keep lots of things open at once and store lots of data without slowness or crashing".
Dual core vs quad, 4GB RAM vs more, CPU architecture are not going to be that big a deal in most general yews situations. Intel Sandy Bridge (Core iX 4-digit) will outperform AMD Llano (A6/A8) in actual benchmarks, but AMD isn't going to be limiting in most situations. If you're considering an AMD and have concerns, we can check benches for specific applications.
______________________________
Recommendations (similar items grouped together):
Acer 15", i5 SB (SB = Sandy Bridge hereafter) dual core, Geforce 540M, $679
http://www.newegg.com/Product/Product.a ... 6834215119Acer 17", i5 SB dual core, Radeon 6650M, $679
http://www.newegg.com/Product/Product.a ... 6834215122For the AMD fan,
HP 15", AMD Llano quad core, Radeon 6620G, BD-ROM, $724
http://www.newegg.com/Product/Product.a ... 6834157877At the HP site, a customized Pavilion dv6zqe Llano based system runs $734 with quad core, 6645M graphics (hybrid crossfire with APU/GPU), 15" screen, 7200RPM HD, and BD-ROM.
http://www.shopping.hp.com/webapp/shopp ... zqe_seriesHP has equivalent Intel systems at 15" and 17" as well.
At the Dell outlet, an XPS 15 (L502x) with dual core i3, Geforce 525M, and 7200RPM HD runs $669.
http://outlet.us.dell.com/ARBOnlineSale ... 1&fid=2149XPS 17 (L702x) for 17" screen starts at $729. They have better graphics but more pixels to push so it's probably a wash in terms of performance.
http://outlet.us.dell.com/ARBOnlineSale ... 1&fid=6589__________________________
Summary Recommendation:
Hang out around the Dell Outlet and pick up a refurb XPS 15 that you like. I've had good success with 2 refurb Dells; Turky has one also. Specific models will come and go by the hour so if you like it, grab it. Dual core Sandy Bridge Core i3 or i5, Geforce 500 series, 7200RPM HD, maybe a BD-ROM, higher end wireless card or bigger battery if you're into that kind of thing. 4GB RAM seems standard, nowadays $50 will get you 8GB if you feel the need to upgrade later, but it's a poor spec by which to choose systems due to the easy upgradability. You should be out the door in the $700-750 range. If 17" is a priority, look at the XPS 17 for $100 more, or compromise drive speed for the cheaper Acer above.