Basically, you need a bobbin lace pillow, some bobbins (at least 24), thread, pins, and a decent book.  Unfortunately lacemakers are few and far between.  If you are lucky enough to have a lace guild in your area, glom onto them right quick.  I used to belong to the Finger Lakes Lace Guild, which meets in Ithaca, NY.

At the website below the basic kit runs you about 60 bucks, which is pretty good.  I started with loaner stuff then bought myself a $200 lace pillow.  

Sadly, this is one of the rare crafts where you can't just learn for free, at least not past the beginner level.  Lacemakers are rare and often flown in from Europe to teach workshops, so to get access to advanced skills you have to fork over a lot of money.  I don't have a lot of money, so I moved from beginner skills (Torchon lace) to what is called free lace.   In free lace, you don't have a set plan of exactly where each thread and pin will go.  Instead you make a general outline, like a sketch, and fill in with what you like.  I saw a picture in an old library book of a spectacular Russian city done in free lace.