A bra is anly a small garment but it needs a lot of different fabrics, elastics and hardware. Buying a kit is one way of getting all (or most!) of the bits needed but in time you will amass a stash of some things and need to stock up on other bits.
Drawing of bra anatomy attached so that it's clear which bit I'm referring to.
Sometimes the wing and the outer part of the cradle are cut as one, sometimes there is a dividing seam.
Cups have at least two pieces, often three or more pieces.
Fabrics:
Generally it is recomended that knit fabrics are used for bras rather than woven fabrics. IMO that is probably as much to do with the non-fray properties of most knits and the narrow seam allowances (1/4"- 6mm) used.
For comfort in wearing the wings need to be stretchy, but at the same time quite strong as much of the support actually comes from the wings. Firm/heavy powernet is best, or a heavyweight lycra but if in doubt use your lycra fabric double.
The bridge, and the rest of the cradle if not cut with the wing piece, needs to be rigid; ie non-stretch. The fabric often recommended is duoplex. Other fabrics will need to be underlined with a non-stretch 15 denier 'bra lining'. Plain net curtain fabric will also work so long as it doesn't fray.
The lower cup also should be non-stretch and underlined with 'bra lining.
The upper cup can have a little stretch in smaller sizes, but larger sizes should be underlined.
If you want to use lace for the cups use skin coloured bra lining.
Elastics:
The lower band elastic is the most important. It is plush on one side and usually with a picot edge.
The under arm elastic is similar, but usually a bit narrower
Some bras have a neckline elastic which is narrow with a fancy edge. Often it's the same sort of elastic which is used around knicker legs. If the upper cup is lace it's more usual to sew a very narrow (2mm if you can get it) soft elastic, or transparent swimwear elastic, along the inside of the cup just below the dips of the lace scallops.
Strap elastic, also sold as suspender elastic, is quite firm with only a little stretch. It comes in a range of widths from about 8mm up to 25mm. The bigger the bra/cups and the more support that needed the wider the strap elastic should be.
Wire channelling is a fabric tube which is stitched around seam between the cups and the cradle and into which the wires are inserted. It is usually a plush fabric but occasionally smooth knit. A non-wire style bra still needs either channelling or a seam tape stitched over the seams.
Hardware:
Hook and eye fastenings are the usual back closure. They can be 1, 2, 3 or more hooks deep and usually have two or three rows of eyes for adjustment. The hooks are always attached the right back and the eyes to the left back.
For a front fastening bra either a bikini clip or hooks with a single row of hooks can be used. The difficulty is fitting it into the width of the bridge.
The straps are usually made with one end passing through a ring and then to an adjustable slider. The ring is attached to the back of the wing, usually by means of a short piece of strap elastic. The diameter of rings and sliders should match the width of the strap elastic. Wider straps use a second pair of sliders in place of the rings.
Wires:
There are a number of different styles of wire depending on the style of bra and it is important to but the right style and the right size. Here is a lot of useful information about bra wires.
[http://sewingchest.co.uk/index.php?cPath=91_37_54]