This is really all you need: A shuttle and a ball of thread!
There are a few tutorials and how-to videos out there, but Jen's Tat-ra Sutra was the easiest to follow, in my opinion, with lots of pictures and easy instructions.
I made my headband based on this pattern, and I'm really happy with how it turned out! Here's how it looked with my final outfit, paired with a flower hairclip (a combination of knitting and crochet):
Are there any other lolita tatters out there? I've been looking through patterns online for lace designs that I could incorporate into lolita and would love to have more links!