Step 1: Where to Buy
But even if your teacher instructs you, you will need to find a cheerful music shop that you trust. You are NOT looking for a shop full of electric guitars here: they need to have at least some expertise with stringed instruments. They needn't be slap next door, but you should be able to reach them relatively easily.
You are going to need strings, sheet music, stands, repairs (or at least setup), bow rehairs, and so forth. OK, you won't need these things every week, but this will be an ongoing relationship. So find a suitable retailer and get to know them.
Then go there, tell them how much you want to spend, and ask their advice. They know that if they make you happy, you'll be back for all the things I listed above - and in a few years, a better violin. And a few years later an even better one. And a new bow. And....
It's in their interest to satisfy you, and they know more about what student instruments are available than you or I ever will.
Step 2: Fab Instruments From Ebay. or Not.
Here are some violins available on Ebay at the time of writing. Do you feel competent to fix them? And these are the ones where you can actually SEE the problems!
* One exception I can recommend: ELIDA TRADING are expert and trustworthy.
Step 3: How Much Should You Spend?
Watch out for the difference between buying a violin and a violin outfit. A violin means just that; a violin outfit will include a bow and a case, which if bought separately could double your outlay.
There's a kind of geometric progression that applies to most musical instruments - and indeed other things like cars. All other things being equal, a $200 violin will be twice as good as a $100 violin. A $1000 violin will be considerably better than a $500 violin. But the difference between a $50,000 violin and a $100,000 violin will be imperceptible to the ordinary person.*
So budget to buy a violin outfit at double the cheapest price you have found - ie as of now, GBP 120 / US $200. Spend more if you can.
I must stress that this is for a beginner: advanced players will not be content with a violin of this standard. But then an advanced player will know all this already! :-)
* This principle is not infallible. It ignores the fact that two $200 violins will be different from each other! But it helps.
Hope you find this useful.