If Photoshop was bundled with your computer, then the serial number would be with the papers that came with the computer.
No, nobody else can legally give you another serial number.
It's probably Photoshop Elements, not Photoshop. Either way you are out of luck if you didn't save your serial number when you bought the computer. But you can "use your pictures" without it. If you need photo editing software, there are free/inexpensive options, just do a search.
I am a little bit confused exactly which Adobe program you have.
Do you have the full version of Adobe Photoshop (not commonly OEM with new computers), Adobe Photoshop Elements (similar, but smaller program), or, are you referring to perhaps Adobe Photoshop Album, or maybe an Adobe Photoshop.COM account, where you store your photographs online, and can have them printed, shared, or similar?
All are different.
One thing that one should ALWAYS do is to Register their Adobe software, immediately. Even if their version does not require Activation (note: that is different than Registering), Registration will put their S/N into their Adobe Account, for later retrieval, if necessary.
Besides Activation (where required), and Registration, I always wrote my S/N into the manual (don't see those any more... ) and usually made a few sticky labels with that S/N, to place on all pertinent disc cases, etc.
Good luck, and please let us know a bit more, as I do not believe that I am the only one here, who is confused on exactly what you DO have.