You have a two part question there so I will answer them one at a time.
You will want to use a RGB profile for any web graphics as that is the standard and it supports a larger color spectrum and all monitors use a RGB profile not CMYK
A good general color profile for hi resolution printing is one of the following two
Working CMYK US Coated
these profiles will be sufficient
I use - FOGRA27 and FORGA39 - for my grand format prints but this is because I am very familiar with the equipment the printer is using
For print I would suggest converting any RGB profiles over to CMYK so you arent shocked as to color shift and brightness of the final print.
for viewing on the web (and on the monitor) -- i would CONVERT the .pdf only to the sRGB ICC Profile and be sure to embed the profile
for printing -- you don't say if it's a CMYK printing press, an ink-jet printer, or if it is going through RIP -- so it would be rather meaningless for me to speculate a profile (use an RGB source profile for most inkjets; a CMYK source profile for offset press; a RIP on an inkjet can print either CMYK or RGB... know your workflow)
i generally keep the original source profile until i make the one Conversion to final destination profile for packaging
read up on Photoshop "Soft Proofing" gamut issues for out of gamut color...