I buy C**h's (as in paper money) woven labels for my Dad and machine-stitch them in; ordered online they arrive quickly. I know others have successfully used iron-on ones too.
But ----- which ever you use, don't be afraid to make them BIG, BOLD and place them somewhere very visible. I used to avoid sewing them on the inside of Dad's shirt collars in case they scratched, but found that putting them anywhere else meant that the laundry staff just didn't see the name tapes, (and couldn't be bothered to look.)
Label everything. Don't assume that things will stay in the room. Take nothing in that you would not wish to lose forever. Mark ornaments, books, CD's etc. with permanent marker pen (and check that it IS permanent!) Try to only send copies of precious photos, and keep the originals safe.
Socks are tricky. Light-coloured ones can be done with marker pen -- I stitched the woven ones onto individual socks, as a kind of loop at the top of the sock, eighteen or twenty socks at a time, but this was no guarantee that there would be a matching pair (or any pair) in his drawer when I visited.
Sorry if this seems negative, but disappearing clothing is a recurring problem in so many otherwise good care homes, and no matter what anyone says to the contrary, it is not fair on the residents, and frustratingly time-consuming for relatives to sort out.
After all this, I do hope your dad settles well, and that the laundry is super-efficient!