Are there any concerns in terms of compatibility with graphics cards and motherboards?
Nope. Any board that will accommodate a GTX 660 will take current cards.
What type of power supply do you have?
I do believe I have the extra 1cm of room for however much longer one of the newer graphics cards are.
Card sizes haven't changed much since your PC was current. If anything they've gotten a little smaller on average.
One thing you might need to watch out for is the monitor connector type. If your monitors are older they could be DVI only.
New cards vary widely as to whether they have a DVI port on them or not. Quite a few do not but they usually have HDMI,
in which case an HDMI to DVI cable will do the job.
There are also adapters but that can be annoying. Very few if any cards will have two DVI connectors. So the 1 DVI + 1 HDMI setup would work.
If your monitors support display port you could switch to that as most cards have 2+ DP outputs.
And for solid state drives, should I still use my current hard drive too? What should I be transferring over if I go this route?
Since you have an older motherboard you'll be limited to SATA SSDs. Which is fine they aren't that much slower anyway.
This is the way I would tackle it:
1. Unplug your current HDD.
2. Install Windows cleanly on the new SSD, make sure it's at least a 256GB SSD. 500+ is better but 256 is pretty workable if you don't have the budget for more.
3. Plug your old HDD back in.
4. In the BIOS set your SSD to be the primary boot device.
That will leave you with 2 working copies of Windows, one on each drive. You could remove the install on the HDD but I usually like having a backup and it doesn't take much space anyway.
Then you'll have your new clean install of windows on the SSD and all your old files intact on the HDD to access when you need them. Usually the HDD acts as secondary storage for things where speed isn't particularly important like videos/music/documents/etc. On the smaller SSD you keep your current games, web browser, and anything else that would benefit from the faster drive.
The other way to handle it would be to clone the drive, which will create a 1:1 copy of your HDD on the SSD. I usually don't like this process for a number of reasons but perhaps the most annoying is that the SSD is almost certainly going to be smaller than the HDD, so cloning a HDD to a SSD can be difficult if that's the case. Some programs (usually included with the SSD you buy but not always) will resize the partitions so they will fit on the smaller SSD. But that only works if you have empty space on the HDD to shrink.