The archaeology field or project director usually gives you a list of what to bring. Most projects expect you to have a field pack for your own belongings, hand tools, and notebook.
All the stuff you bring on a dig stands a good chance of getting damaged or destroyed, so start with a good, sturdy — but perhaps not new — pack. Used military packs of heavy cotton canvas are great, though you may consider buying a ballistic-nylon bag in a bright color so you can easily locate it.
Ask what hand tools are supplied or whether you need to buy some for yourself. Here are typical excavation tools in an archaeological fieldworker's pack:
Pointed 4-inch Marshalltown trowel (and maybe a square one, too)
3-meter measuring tape (harder to find than inches-and-feet increments) or other small tape
Butter knife, grapefruit or regular spoon, and pocketknife
Artist's spatula and dental pick (ask your dentist for old or broken ones)
Sharpened wooden chopstick or length of bamboo for softer digging
Small, clean (but cheap) paintbrush for brushing away dirt
Roll of plastic flagging in a bright color
Water-resistant field notebook (check dig requirements)
Compass (cheap or expensive)
Pencils (and a sharpener), waterproof markers, waterproof pens, and/or space pen
Line level for measuring depths from a level string
Zipper-lock plastic bags of all sizes for finds