If you're new to the sport, do yourself a favor and invest in the following. You won't regret it.
Nippers – Quality makes a difference for cutting line with ease. Some nippers also come with a nail knot tool.
Knot Tools – Nail knot tools make line repairs and permanent knot tying easy. Keep one handy at all times.
Forceps – Quality forceps or mitten clamps are useful for pinching barbs and removing hooks.
Floatant – Paste, spray and dry shake are all common options.
Fly Boxes – Organization is important. Carry several boxes to separate your dry flies, nymphs and streamers.
Split Shot – Use a non-toxic brand with multiple size options.
Strike Indicators – Yarn, cork and plastic indicators are popular. Keep several sizes in your pack.
Leaders and Tippet – Always carry several extra leaders and spools of tippet. You will need them for fly changes throughout the day.
Flashlight – A small flashlight or headlamp is useful for fishing late in the evening. It will lead you to the truck when the light is running out.
Bug Repellent and Sunscreen – You will be happy to have them..
Camera – If you prefer to leave the phone in the car but still want to get some great shots, do yourself a favor and go with a waterproof camera if possible. Otherwise put unprotected models in a plastic bag to prevent the inevitable.