To get the scan to your new machine a USB thumb drive would be the simplest way. Scan on the old machine and transfer to the new. Or if you have a CD-RW in both machines you could format a RW using packet writing to use it like a big floppy to transfer files. As far as I know scanning does not work over a network anyway.
To share the printer you have to network your computers. A crossover cable would be the easiest/cheapest(about $10) way to connect the computers since you don't need internet access on the old machine.
What I do is I have my machines networked with a router but only one machine setup to share files. I have the printer/scanner hooked up to that machine, and have the scanner set to save to a shared file, then I can get the file from any computer in my house.
As already stated a KVM (Keyboard,Video,Mouse) switch will work for the rest.
Ultimate Edition Linux
AMD FX 8350
Asus Sabertooth 990FX R2.0
24GB GSkill/Corsair 1866MHz
EVGA GTX 970 SSC/ GTX 650 (PhysX)
SSD - OCZ V4 128GB/Kingston 120GB
HDD - 3TB/2x 750GB/500GB
CM HAF 932 Red
CM Hyper 212 EVO w/2x 120mm Red
Logitech X540 5.1
2X Acer 23" LED - Surround: 3840x1080
Microsoft Sidewinder x4 KB
"The significant problems we face cannot be solved at the same level of thinking we were at when we created them."
- Albert Einstein