As a general rule, you can handle and work with photo inkjet prints as soon as the come out of your printer. They are "dry to the touch" but in reality it takes about 24 hours for an inkjet print to fully cure and dry. There are some points you should know about and consider when you are printing.
Wait before deciding on color quality
The colors of a photo print change over the first hour or so of drying. Always allow at least an hour before deciding if you are satisfied with a print.
Going to frame a print? Wait at least 24 hours
If you are going to frame an inkjet print under glass or plastic allow at least 24 hours of drying time. If the print is not fully cured a light haze can appear on your glass. This is called outgassing. The haze is the ink solvent continuing to escape from a print as it dries.
Here's a great article on framing and mounting techniques for your learning pleasure.
Want to try and speed up the process?
Put a sheet of plain copy paper on top of your print when it is complete. You will notice the copy paper begin to curl as it absorbs the escaping ink solvent.
For best results don't stack prints
It will take up more space, but your prints will dry faster if allowed to sit out unstacked. Put a sheet of copy paper on each print (see above) to prevent dust and debris from settling on your work.