You got the hardware pretty much figured out. For video recording you would need an always-on computer or network storage device.
Remote monitoring is a bit tricky. You would need a fixed IP address from your Internet service provider or you could opt for a Dynamic DNS service which eliminates the need for a fixed IP and gives your camera a web address. If you have a Dlink router you can avail their free dynamic DNS service. I have Dynamic DNS set up on my Dlink router so I can remotely connect to my home computer.
Other than that you would need to set up port forwarding on the router to access the camera remotely. For the non-technical person it can seem pretty daunting, but it can be done.
If you want to keep it simple, connect your IP camera to your router. Install Teamviewer on remote PC and also on the home PC. Use Teamviewer to connect to your home pc from the remote location and then you can access the IP camera as if you were sitting at home. But Teamviewer would require a faster internet connection, otherwise you will suffer from a lot of frame lag.