Best practice is to use an anti-static strap, but I personally just use the "touch the case metal". I tend to sit on the floor anyways working on my PC, so I keep a bare foot on the bottom of the tower. LOL
But as a side note, it doesn't take much of a static zap to render a RAM stick or CPU dead, or damaged. And the bad thing, it may not kill the item, just cause it to BSOD, or odd things to happen.
You don't really need a grounding strap at all, I've never used one. ESD is not as big an issue with building pc's as some people make it out to be.
Just stay relatively still while youre touching the components you'll be fine.
Unplug your PC and press that power button once it's unplugged to purge any electricity in the system. The alligator clip just attaches to the case of your computer. I don't use anti static wrist straps, if you just touch the computer case you'll dispose of any static you've gathered. You should be fine as long as you aren't dragging your feet on the carpet before building it. The wood table will be great.