Thank you - that's the reassurance I needed. I've got SSDs but was going to stick those inside the PC, so this is just for some older hard drives. I also just found a 2TB drive I forgot I had so am going to try and do some reorganisation and reduce the overall number of drives.
Pretty much any PCIe eSATA card will do.
SATA 3 supports faster transfer than SATA 2. This is academic for a standard HDD but may make a difference if you were to move to an SSD.
You could also consider a USB3 to eSATA cable, or a new USB3 enclosure (assuming it's a standard HDD inside).
The top slot looks like a PCIe x16 slot, so it'll take any PCIe card, and the second one down a legacy PCI slot - ignore that one.