The 5 accessories you actually need
The accessory wall at a well-stocked fly shop is overwhelming on purpose: clips, retractors, drying patches, magnetic net holders, leader straighteners, and a dozen devices whose function is unclear until someone explains it. Most of it works, and almost none of it matters. A beginner needs five things on the water, and once you have them, you are done with the accessory question for years.
All prices below are in USD.
1. Nippers

A small pair of clippers that cuts tippet cleanly after you tie a knot, which is something you will do more than any other action in your kit. Your teeth are not the answer, because they will eventually fail you and tippet is hard on enamel.
What to spend: $8 to $20. A basic stainless or aluminum nipper from a fly shop or hardware store cuts tippet exactly as well as a $75 anodized titanium version. The expensive ones are lighter, prettier, and sometimes come on a retractor, but they do not cut better.
2. Hemostats

Surgical forceps with a locking jaw, originally a medical tool. Fly fishers adopted them because they do four things well: hold a small fly while you tie a knot, pinch the barb down on a hook, remove a hook from a fish, and remove a hook from yourself when a back-cast goes sideways. The fourth happens more than beginners expect, so crimp the barb on every fly before you tie it on. A barbless hook backs out of skin without a fight.
What to spend: $10 to $25. The cheap pairs at the fly shop counter are fine, with stainless steel, locking jaws, and scissor handles. A $50 "premium" pair gets you a logo and a nicer finish.
3. Polarized sunglasses

This is the one accessory that changes what you can do on the water, not just how comfortably you do it. Polarization filters out the glare bouncing off the water's surface, and a river that looked like a moving mirror starts to look like a window: you see the bottom, you see where the current breaks around a rock, and you see fish holding in lies you would otherwise walk past.
What to spend: $50 to $100 for glasses that polarize correctly and have lenses that will not scratch in your first season. The $350 pairs from outdoor brands polarize the same way, with the extra money going to better hinges, lighter glass lenses, and brand cachet.
The one spec worth caring about is lens colour. Amber, copper, or brown lenses brighten contrast in the kind of low-to-medium light you will fish most often, while grey lenses cut brightness more aggressively and are better in open sun on saltwater flats. For freshwater trout, start with amber or copper.
If you wear prescription glasses, try clip-on polarized fitovers before you spend on prescription polarized lenses. Fitovers are $30 to $50 and work surprisingly well.
4. Floatant

Only relevant if you are fishing dry flies, which most beginners will be at least some of the time. A dry fly is designed to float, but after a few drifts it absorbs water and sits lower in the surface film. Floatant solves that with a small dab worked into the hackle and wing, after which the fly rides high again.
Floatant comes in two formats: silicone paste in a small bottle, dabbed on a fingertip and rubbed into the fly, and liquid floatant in a dip bottle, into which you dip the fly, shake off the excess, and watch it float hard for the rest of the session.
What to spend: $10 to $15 for a small bottle, which lasts most of a season. Skip powder floatants and performance gels that cost three times as much. They solve problems that are not yours yet.
5. Strike indicators
Only relevant if you are fishing nymphs (flies that drift below the surface, where most trout do most of their feeding). A strike indicator is a small bobber on your leader above the fly that twitches or dips when a fish takes the nymph below, and without one a beginner has almost no chance of detecting subsurface strikes.
Four types exist (foam bobbers, cork sliders, yarn indicators, and putty that molds onto the leader) and they all work, but the foam ones are the easiest to see and the easiest to attach.
What to spend: $10 to $15 for a pack that lasts most of a season. Skip anything marketed as a "system" or requiring a special leader configuration. The simple foam bobber is the right starting point.
What we deliberately left off this list
A lot of accessories that show up in beginner kits do not belong there yet.
Nets. Useful eventually, especially for catch-and-release on bigger fish, but you can land small trout by hand in shallow water. Buy a net when handling fish without one starts to feel rough.
Chest packs, sling packs, vests. Storage problems before they are tool problems. A jacket with two pockets holds your nippers, hemostats, floatant, indicators, and a leader spool, which is the whole kit.
Lanyards, retractors, zingers. Tool-management products for a problem you do not yet have. Hemostats clip to a shirt placket and nippers go in a pocket, and you can spend $100 organizing tools you barely use.
Gadgets. Knot-tying tools, leader straighteners, line cleaners, hook hones, drying patches with elastic clips. Each solves a problem that does not really exist for a beginner. Wait until you have felt the problem yourself before you buy the cure.
The kit, in one sentence
Nippers, hemostats, polarized sunglasses, a small bottle of floatant, and a pack of indicators. Five items, $90 to $200 USD depending on where you land on the sunglasses, and they fit in a jacket pocket or a small chest pack with room left over for tippet spools and a fly box.