Better tell us what kind of fly fishing you'll be doing.
but you will need
Waterproofs and hat (hat may help in keeping off glare but protects your noddle from ill directed flies - remember, they can be travelling at 100mph plus)
Glasses of some kind to protect eyes from flies (polarised sunglasses can be very useful under some conditions)
Fishing waistcoat (vest) if you like the idea but a jacket with voluminous pockets also works
Waterproof footwear - even if fishing stillwaters it can get muddy.
Waders may or may not be needed depending on where you're going to fish.
A fishing bag is useful, even if only for your thermos and butties. A rucksack will also do.
A flybox (or several)
A priest if you intend to take fish.
Floatant (eg Gink)
Sinkant (eg Leedasink but you can make up your own if you want) Holders for these can be bought for your vest
Scissors and/or snips (fingernail clippers will do)
Forceps for hook removal
A zinger to hang it all from can be useful
A sewing needle for clearing out varnish from the eyes of flies
To begin with some tapered leaders (9ft is probably an adequate allround size)
Some tippet material - size/type will depend on where you are fishing and for what.
Midge repellant - worst is well over now.
Suncream
Landing net
Amadou is good for drying flies, but a piece of genuine chamois will also do the job.
Book of good excuses for your missus
a) for all the money you are about to spend on a highly addictive pastime
b) for persistently sneaking off fishing and arriving home long after dark
There is no end to the gear that you can acquire, nobody ever has enough, but really there is a lot to be said for keeping it simple.
Just seen you'll be fishing rivers and saltwater so sinkant probably not necessary . If you are going to wade, a wading staff and preferably a life jacket of some kind. A chest bag can be very useful when wading.