An SSD will most certainly benefit moving from SATA 2 to SATA 3.
A hard drive not so much. My WD Black 1TB, one of the higher performance hard drives, can write at about 150MB/s sequential. That's half of SATA 2. And my SSD will write about 515MB/s. But that's not the full story. There's much to learn about SATA and storage drive speeds and would certainly be worth the effort to learn.
For quick reference. SATA 1 is 150MB/s, SATA 2 300MB/s, SATA 3 600MB/s. Sequential writes and reads are slower and apply to larger files like movies. Random writes and reads favor smaller files and SSDs really excel at that.
PCI-E 3.0 has 985MB/s data throughput per lane. PCI-E has a max size of 16 lanes. Motherboards will support x1, x4, x8, and x16. The amount of lanes used will depend on the expansion card. Since you mention you have no SATA 3 it's likely you have PCI-E 2.0 in which case the max data rate is 500MB/s per lane.