Tools and Materials
I guess the first thing I should discuss would be the tools and materials you will need.
Luckily the most expensive thing is a nice sharp pair of embroidery scissors with pointy tips.
These are much more necessary for cutwork than for punto-in-aria for obvious reasons....with cutwork many times you are needing to snip just a single thread of the cloth before pulling it out and you need the most precision you can get. These are Dovo's from Solingen,Germany, given to me by my wonderful husband. I love them...they don't leave the house.
The next necessary tools are your needles. I use both John James size 11 sharp short beading needles and Dritz size 10 ball points. I've tried size 12 sharps and they start getting too flexible to grip properly. The holes on the sharps are rather small so you may have to go up a bit in size if you are using thicker thread.
Some of you younger folk might not need these but since I turned 40 I cannot get by without reading glasses as well. The doc says I need 1.5's but for lace work I kick it up to 4.0's. and I found ones on Amazon that have little LED's built in that I am giddy about...they may help in the car! Which leads to lighting. a good but diffuse work light will help tremendously.
For Cutwork I use a 4" spring tension embroidery hoop which allows my fingers to easily get to both sides. Not period, but much more ergonomic and portable than a slate frame.
For Punto-in-aria the tension is kept by the paper pattern. In period they backed them with fabric for strength, but I use upholstery vinyl over my paper patterns to keep the printer ink from rubbing off onto my work. (learned THAT one the hard way) they sell the upholstery vinyl in JoAnne's fabrics and a yard will last you for years.
A paper pattern on fabric partially worked (from slightly after my period of interest) is in the Cleveland Museum of art!
Another thing that you will need for cutwork is Starch. I use wheat starch and I bought a lb many years ago and keep it in the freezer. I mostly use only 1/8th of a teaspoon at a time so it will probably still be there when I keel over. (unless I start making ruffs and stiffening them!) To mix it up I put 1/8th of a tsp in a sake cup with a tsp of water and microwave it for about 15 seconds. You want it to be the consistency of snot when you apply it. It really helps keep the pesky threads in place when you are doing the overcast edging. I apply it with a tiny paint brush or a q-tip in a pinch.
Lastly, for Punto-in-aria, a pair of tweezers or small jeweler's pliers is helpful for removing the tacking stitches at the end when some of them are being stubborn.
Oh, and the linen thread and fabric...don't want to forget those!
They all have slubs. I haven't found a single one that doesn't. but some seem to be a bit more consistent. I've tried and liked pella, goldschild, bockens, and freesia. I tend to use 100/2 for my most delicate work but it does break a lot. I would recommend starting with something slightly bigger and less frustrating. I start my students with 50/2 so that I can see what they are doing...and let them then move on their own to 100/3 or 80/2 perhaps? here in the States, Greylines linen up in NYC has a couple of decent fine weight linen fabrics. but my latest find is Sartor's fine weight linen...very nice! and almost an even weave!
Overall, needle lace is rather economical hobby. especially when compared to goldwork!