I recently bought a new desktop PC. I'd like to be able to access my old desktop PC from my new one. Not just access the files on the old one, but control it and run programs. I have some old programs that I no longer have license info for that I'd like to continue to use.

I don't really have a home network per se, just a router.

I know my new desktop is running Windows 10 home, and I think my old desktop is home too.

Is there an easy way to do this, or should I get a KVM switch?

Couple of options.
1. KVM
2. Remote Desktop where you remote control the other computer. Not really good at super high refresh rates.
3. P2V conversation where you make the old machine into a virtual machine. Might cause problems with licensed programs.

What kind of software is it?

If it's something like video editing software, then I would use a KVM switch.

If it's something like accounting software in which you can tolerate crappier refresh rates, you could just connect the old PC to your router (use some CAT5/6/whatever; don't use wifi for this) and run it headless with something like TightVNC.

Mostly I want to use a copy of Adobe Acrobat Pro X to process large document sets. Flatten, OCR, produce TIFFs, Bate numbering, etc. Nothing that requires high FPS. I have Nitro Pro on my new computer, but it just doesn't do somethings as well, and some of the processes take a long time and slow down work so I want to run them on my other PC.

I also want to be able to access and move files between computers.

Is this something I can do with TightVNC?  Can I completely close the connection from the outside world to reduce risk?

It sounds like TightVNC would probably work well for what you want to do. It's certainly not as "smooth" as having an actual KVM connected, but for "this and that" it's probably fine. If you were talking about using the software for hours on end, you might grow weary of it.

Might as well give it a shot. TightVNC is free and only takes a few minutes to install.

TightVNC will let you remotely control the other PC. You technically can copy/paste files between the two PCs, but that would be an asinine way to move files around.

The better way to do it would be to a) "share" the drive that houses the files (Google something like "Windows share Hard drive" and I am sure you will find tons of guides; it's simple) or b) store the the files in "the cloud" with something like Dropbox or Google drive. B is probably the better option since 1) the data is probably more secure on in a datacenter than your computer's hard drive and 2) you can access inside our outside your LAN.

Remote Desktop

To just access files:

To run programs on the old computer, use Remote Desktop.

Windows 10 Home doesn't have RDP server enabled, only client. You'll need to upgrade to Pro or Enterprise. Alternatively, there are some hacks to get the server running since the server components are installed but not enabled by default via license (search for RDP wrapper for more info). VNC might be a better alternative, and either should be behind stout firewall to block out external traffic.

Cheap way... cross-over Ethernet cable.

I use to do this for my old laptop. The hdd wasn’t big enough to hold all my music so I used crossover cable and played the music from the desktop instead.

You may need to configure the internet settings a little though. I don’t really remember what I did. This was over 10 years ago.