Starting a Wardrobe/Fashion Stylist Kit is not hard or too expensive to do- if you know where and when to shop - (shop when they are on Sale!). Some of the items are "no-brainers", but some I will let you know what I use them for. I will also place links if they're items you might not know where to get them. Keep in mind that this will be Part 1 of 5...
Here are a few items that I have had in my kit since I started and that I use all the time:
Kit Bag - this is where I keep everything
Clamps - we use these for making things look fitted on photo shoots. They are about $7 for a mixed pack of 22pc
Top Stick - we use for keeping "things" in place and no wardrobe mishaps on the Red Carpet
Swiss Army Knife - you will find just how often you will use this!
Fabric Scissors - these are one of the ones I have (I have 5 scissors in my basic kit)
OxiClean Spray-A-Way
Shoe Horn - any size will work - you might want to get 2 - they tend to disappear on set
Static Guard
Mini Sewing Kit
Wet Ones
Scotch & Bounce Lint Rollers
Basting/Safety Pins
Ziploc Easy Zipper Bags - get a few sizes - they help you organize items on set and in your kit
With these items you should be good for a basic shoot - but keep in mind that there really isn't such thing as a basic shoot. ;)