Q: What's the simplest way to make my computer at home accessible to a computer elsewhere (with password protection and all that) like my work PC or my laptop?

A: Simplest way is with LogMeIn free. You just need to register and install the appropriate extensions/plug-ins on the home PC. Then on any remote PC, go to logmein.com, login to your account, and you'll see the computer ready to be controlled. You can initiate full remote control, or just file transfer session (i.e. grab a file you left at home).

Correction: apparently the FREE version only lets you remote control, NOT file transfer. The "Pro" version, which you do get a 15-day trial, lets you do both, but costs money.

A truly free alternative is Team Viewer, which is free for personal use. Install it at home, and it'll generate a self-ID and a password. The other computer also installs Team Viewer. I'm using it to control my home PC now, and transfer files. It works great.