some disconnected thoughts (and partly copypasta from another thread a year or two back)
this page on government website may be worth a look for the tax side of things.
i had a brief foray in to self employment alongside part time employed work a few years back, and while it means you get in to the realms of tax returns, they didn't ask to see a lot of paperwork etc (my self employed income was fairly small and they probably didn't think it was worth it. i'm not sure it was worth it from where i was sitting, but it looked better than 'unemployed'...)
without knowing what you do, and at the risk of stating the obvious -
your taxable income from self employed work is the net income / profit, not the gross income. Although you'll need to keep records of costs. Where it's buy stuff for X amount, sell it for Y amount, that's relatively simple. Costs like business travel, stationery, advertising etc are all legitimate costs to deduct. Will you need to provide any computer hardware / software for whatever you do?
if you're going to do stuff as a contractor not an employee, you should probably be charging more than an hourly wage would be - you're taking on responsibility for holiday pay / sick pay / NI contributions / the time it will take to do accounts and bill them, in addition to the hours you're actually working. when i worked in the consultancy field (albeit as a permanent full time employee of consultancy company) what they billed the client was in general about 2x what they paid me.
will you be doing whatever it is from home? May be worth reading small print on mortgage / tenancy agreement to make sure you can - also could have implications on house insurance and council tax in a way that (employed) work from home doesn't - but if you do enough from home then you might be able to claim some of your household bills as business expenses.
And do you need to consider taking out any sort of public liability insurance, or professional indemnity insurance? if you're an employee and you balls something up, you run the risk of getting sacked. if you're self employed and balls something up, you run the risk of getting sued.
setting up as a company is way beyond me.
i'd agree that it may be worth asking if they would consider you doing it as a zero hours contract (or contracted for something like one day a year with the option to do more hours) employee rather than get in to all this faffing about if it's only going to be a couple of days' work. depends how much you want to do it / how much you'll make from it / how much other self employed work you might do.
i have heard of the concept of doing self employed work via an 'umbrella company' but not sure how this works or if it might work for you.