I have seen articles and youtube videos about accessing the embedded systems on various electronics such as routers and digital cameras.
Here are two inspirations:
http://www.devttys0.com/2012/11/reverse-engineering-serial-ports/

I sort of follow how to find the serial port on the board but I don't know how to do the rest, such as which serial connection cable to buy and what program to run on my computer that will monitor the serial port. If you can let me know that would be great.

What would be nice is guide to hack some easy device, lets say an old router that can be purchased for cheap on ebay and a step by step instruction for a beginner to read the serial port and communicate with the linux running on it.
Then I can use that to get into other more complicated devices.

 

Older Windows had HyperTerminal, it was program that worked with computer serial ports. Now that most manufacturers stopped putting serial ports on the computers, M$ stopped including this program.

If you have Win 98/2000/Xp, you can use HyperTerminal. It should be in Accessories.
Or you can use something else, just Google: serial port monitor.

 

I still use ProcommPlus V4.8 on Win7-64 but that's getting hard to find.
Try Tera Term. Some clients use it and its seems to be pretty good. I think it has a MONITOR window like Procomm that shows incoming and outgoing serial data.
Good luck.

 

Older Windows had HyperTerminal, it was program that worked with computer serial ports. Now that most manufacturers stopped putting serial ports on the computers, M$ stopped including this program.

If you have Win 98/2000/Xp, you can use HyperTerminal. It should be in Accessories.
Or you can use something else, just Google: serial port monitor.

You can still download HyperTerminal. I did this a couple years ago and still use it. I also use an FTDI USB-to-Serial converter, which appears on my computer just like an old serial port. Just use the port number in HyperTerminal and you're on your way.

The USB-to-Serial converter I use doesn't have RS-232 level signals. For that, all you need is a USB-to-RS232 converter.