Tony, if you get a USB 2.0 "printer" cable (standard USB Type-A-to-Type-B cable) then you can connect your DGX-660 to your computer. You may or may not need to install the appropriate Yamaha USB-MIDI driver on your computer; I believe the DGX-660 is USB-MIDI class-compliant, but even so it's sometimes necessary or recommended to install the driver.
Virtual instrument plug-ins are available all over the internet, ranging in price from completely free to several hundreds of dollars. As is so often the case, "you get what you pay for," which is not to say that there are no decent free plug-ins (because there are) or that all expensive plug-ins are worth their price tags (because some of them aren't). But generally speaking you'll probably need to spend some money-- possibly a lot-- to get truly great virtual piano software, although you might want to start with free plug-ins to see what's what.
Aside from a USB cable, the USB-MIDI driver, and some virtual instrument plug-ins, you'll need some software to host the plug-ins. Typically a DAW (or "digital audio workstation") will be used for this, since it can also be used for recording and editing audio and MIDI tracks, mixing them together, and producing a final audio file. DAWs range in price from free to expensive, so you might want to start with a free one, but the free ones often have much more limited features. And one thing that tends to run up the price of professional DAWs is the inclusion of virtual instruments and effects, so you might want to look at the list of virtual instruments that are included, since you might be able to "kill two birds with one stone" by choosing the right DAW. However, if you'd rather not get into DAWs right now, there are also simpler plug-in hosts available for free-- just search for "free VST host" or something like that. (By the way, "VST" isn't really the best term, because it refers to a specific type of plug-in technology, and there are also other types-- e.g., "AU" and "AAX"-- so the best term when you're speaking of plug-ins in general is "virtual instruments," or "virtual effects" if you're talking about effects plug-ins.)
EDIT: PS-- One thing that can have a big impact on the quality of a virtual instrument is whether it uses sampled sound or modeled sound. Sound samples are fine-- that's what the DGX-660 uses, after all-- but there are limitations to how much tweaking can be applied to it, not to mention problems with getting a really good sustained note. On the other hand, sound modeling is more expensive but can produce better-sounding results if it's done well. That's why software such as Pianoteq uses sound modeling-- and why it costs more than a virtual piano plug-in that merely uses sound samples.
