The question has come up a number of times recently and often does....I'm gonna start this thread and hopefully keep adding to it for a while....when I have the time...I will post a reasonable list of starter materials,tools etc. for the beginner as well as start posting some very easy beginner patterns that will catch fish...and advance from there......
One thing to consider before jumping in is how serious are you about tying flies....there are two types of tyers in my opinion....the guy who is gonna tie a few flies here and there just to supplement his box....
and those who are going to truly make a hobby of it.
If you are only planning on tying simple flies to fish with...you will need only the basics...if you really wish to learn to tie a whole wide range of flies...you will need advanced tools and a higher quality vise...I recommend getting these from the start!
Inferior tools and materials will only cause frustration and disappointment in your flies.
With that said...I'm going to recommend two vises....
1st the simple version... A "regal" knock-off these can be purchased for as little as $30...I recommend these because other vises that use a lever as a cam break easily...both the pins and the jaws....this vise has very strong jaws,and uses a spring action that when the lever is pushed...the jaws open when released the jaws close securely on the hook...
If your a little more serious...and believe this is something you want to do for the long term...and your going to commit the time and money to tying flies for some time....I recommend the Renzetti cam traveler....just ask anyone who has one...they are the best of the best ...dollar for dollar...there are more expensive vises, there are other "high end" vises slightly cheaper....but this is the one I'm most comfortable recommending...
(I also recommend the pedestal base version) C-clamp model is pictured
Other tools you will need include a bobbin...this is the tool that holds the spool of thread....they can be bought for a couple of dollars...I recommend the ceramic tip version...around $10
a bobbin threader....tool with fine wire assists in getting thread thru the tube of the bobbin.
Scissors.....a cheap pair can be bought for about $5 or I recommend Dr.Slicks...or Anvil ice tempered scissors. These will run $16-$25 or so. get a pair with a fine point...and DO NOT CUT WIRE with them....find a small pair of side cutters or buy a "cheapie" pair of scissors for that.
Anvil Ice tempered...
I also strongly recommend the Matterelli style whip finish tool...there are folks who can put a whip finish on a fly with thier fingers,and are good at it...but I recommend learning to use this tool....(this ties the knot to finish the fly)these tools also come in a "midge" version,and an extended version...the extended version is good for tying behind the heads of poppers...and the midge version is to give more control when tying teeny tiny flies the basic "no name" version $4-$7
a bodkin...which simply put is a long needle with a handle...it has a thousand uses! $2
a pair of tweezers...preferably something with a needle point that opens when you squeeze them and closes when you release them $2
A hair stacker.... small two piece tool for aligning the tips of deer,elk and other hair used to tie flies.Stackers come in several sizes from small to large and in several materials including plastic,brass,aluminum,and wood.Some are made to reduce static...but all I've ever used is the basic brass version $5 -$10
hackle pliers...small device used to grip a hackle feather while wrapping it....again many kinds...my recommendation...."duplex jawed " pliers...one side is metal the other is textured rubber. $3-$5

A dubbing loop "turbo" twister....this is a weighted tool with two "hooks" that is used to spin a dubbing loop into a dubbing rope. $5-$10
these things should give you a really good start...there are many other gadgets that can be bought such as a dubbing teaser,fur comb or rake,hair snare,parachute gallows tool.and so on....but they are not "essential tools.