While I'm hardly a board repair guru, I have done my share of board repairs and I was trained extensively at work on the evils of ESD. My MO is generally to be more careful the newer the board is. I haven't killed a WPC board yet and I don't wear kid gloves (a ground strap) while handling them.

Always use an ESD/anti-static bag when moving or shipping. They are cheap and available online. Pink bags, blister or otherwise, are supposed to be ESD safe. You can now get very nice large pink ESD zip-lock bags, if that's what you prefer. Black ESD bags are fine too, as long as you're sure they are ESD bags. Chips go on anti-static foam, typically black. Then in a ESD bag, if possible. No tape on on ESD bag, just fold it over. The tape can generate it's own ESD.

I don't wear a wrist strap, but I avoid carpet as much as possible when handling boards. That would be the most important advice I would suggest to folks learning to work on boards. Stay away from carpet. If you have to carry a board over carpet, bag it first and touch something that's grounded (like the side rails of a plugged in pin) before you remove the board from the bag. ESD (electrostatic discharge) can deliver tens of thousands of volts to your board and chips. That's bad!

wrist strap and ES bags YES

Use metalized static shielding bag for shipping. The pink poly is anti-static meaning that it won't generate static but also won't protect against static either. IC's in conductive foam (puts all legs at same potential) then inside static shielding bag.
Some IC's hold up better to ESD than others. 4000 series (CMOS) IC's don't hold up well and neither do many memories (5101's are notorious for being easily killed). FET type transistors are also quite susceptible. Rectifiers such as bridge rectifiers and 1N400x series - these are fairly impervious to ESD yet should still be treated as an ESD susceptible part.

Jameco sells smaller, affordable ESD static mats - that's what I have on my work bench. Wrist straps can get pricey but watch ebay. You can find good deals on ebay but watch out for old, stretched out wrist straps.