There isn't really a standard size for this kind of thing and even if there was, there are very good creative reasons to do something non-standard.
The crucial thing about save the date cards (and invitations, etc) is that they need to fit in an envelope so your first step would be to find out about envelopes that are available and that you like the look of. You then need to define a size for the card that will fit in the envelope, both in terms of dimensions and thickness. It doesn't have to be square or rectangular, it could have rounded corners, cutouts, etc. At this point it would be a good idea to create a blank sample of the card and the envelope and get it approved by the client (if there is a client). Once you are all agreed on the physical aspects, you can commence with the design.
In terms of software, you could do the whole thing in Illustrator if your image is to be placed as it is or if it only requires minor adjustments (such as opacity), but if you need to apply radical changes to the image then Photoshop is the better route for that part of the project, then continue adding text etc in Illustrator.