Question:
Would it help me in turn times to have a better graphics card?
I have a 5700 ATI (see attached)
Almost certainly not. It would just make the graphics maybe run at a higher frame rate which could make it look slightly better while you were waiting...
An HD 5770 is something like 5 tiers better than the card I use (an Nvidia GT 520, which is probably slightly worse than the GPU built into most of the current Haswell Core i3/5/7 processors, but better than the one built into my Ivy Bridge generation CPU). I can play C2C, although the frame rate stinks even with the graphics turned down some. This has little bearing on the turn times.
Also if i got am SSD would that help even more??
Probably not, since you have enough memory that you should not be using the page file much (if at all) unless you are also running some memory hungry programs at the same time as you run the game. What it would do is make it load the game a lot faster.
The main thing that would help turn time is a faster CPU. The is almost always the case if the system has enough memory. The CPU you have is pretty good - it was about as good as you could do 5 years ago. But time passes... A current generation high-end i5 would be measurably faster for C2C now.
A problem you have is that upgrading just the CPU would be difficult since the socket your CPU and motherboard use (LGA 1156) was not used anymore after that generation of processor (oddly, the next generation apparently went back to the socket used by the previous generation, the LGA 1366, and the two generations after that, the Sandy Bridge and Ivy Bridge generations, had a new socket that is also no longer used by the current Haswell generation). It is hard to get a CPU that fits the LGA 1156 these days - Newegg, for example, only has 1 type available and it is the i5 version of your processor (same clock speed but no hyperthreading).
If you could get one, an i7-880 would be maybe 9% faster, which is not a huge improvement especially for the cost. As far as I know that was the fastest CPU made for the LGA 1156 motherboards. When available they seem to run something like $350 (you can probably find them for around $1000, which is a rip-off), which seems excessive since for $350 you can get a brand new motherboard and a faster CPU of the current generation. Example: a Core i5-4670 goes for about $220 at Newegg (which might not be the lowest price for this, but is pretty good) and is certainly, shall we say, "somewhat faster" than an i7-860 for anything using up to 4 threads. Probably about 35% faster, actually, so turn times would probably be about 25% less. But that is just the CPU - upgrading with parts always requires checking everything since a new motherboard (which would be required) may not have some old connector type you are using for something, like your DVD (last time I upgraded, in Jan 2013, I had to replace a perfectly good DVD-RW since it used a parallel ATA connector and the motherboard I got only has serial ATA connectors for disk drives).
Technology marches on. On the other hand, for single threaded processing purposes CPUs are not getting faster at the rate they used to - the current Haswell generation from Intel is less than 7% faster at the same clock speed and core/thread count than the previous Ivy Bridge generation, and the CPUs in a category are generally clocked either at the same speed or maybe 100 MHz faster. What this means is that upgrading now will not leave you seeing computers that can run C2C 25% faster come out in another few months (when the next set will probably be released - the "Haswell Refresh") - maybe 10%, but probably only 5%.