Begin by backing up your current drive to the one you want to swap in. That way there's no long evenings re-loading Gig's worth of applications and their serial numbers *shudder*
Then, look up the manufacturer of your machine, and look up the support files for your specific machine. Poke around a bit to see if there is a video that explains Hard Drive Swap. If the computer manufacturer doesn't host a short video on it, look for a hard drive vendor that does host hard drive swap info. When you find a vendor that has video instructions I recommend buying from this company. Be fair and support the companies that help you.
Try to find at least two different videos on your subject. Frequently there is better video on one and better spoken instructions on the other. It's just the way it works out *shrug* I dunno.
When you have your video instructions ready watch them repeatedly so that you nearly have it memorized. Make sure you understand any and all terminology and have all the necessary tools. If possible have another computer in a convenient distance (same house is best) so that you can refer to the video if you want. You might also consider downloading the video and burning it to a disc so you don't need to be concerned about connectivity. If possible have someone turn it into a DVD so that you can watch it on a big screen as you work. (As an after thought, you might like to have your cell and the vendors tech service # handy just in case things go really wrong. It's all about being ready for the unexpected.)
Put any four legged friends in another room behind a door, turn off the phones and lower the music volume. It's easier to concentrate then.
Work on a clean towel. This way dropped items don't bounce off the hard surface.
Arrange your tools by kind. All Philips screwdrivers are together, all Torx are together etc
Keep a good supply of Stickey pads or similar next to where the machine is.
Don't spread out all over the place. Keeping things together means they don't get scattered.
Only pick up one tool at a time and when done with it, put it back from where you got it.
NOTICE I live in a humid locale so STATIC isn't an issue for me. You should wear a grounding strap though. They cost about $2 but will save your $2500 laptop from being fried. You can cook electronics with a static charge so small you don't even feel it. It's that old Safe / Sorry thing. Follow the instructions on the package of the strap.