Fairness is a tough one... A fair tax system also might not necessarily be the best system in terms of effect on business etc...
But, What I think would be fair is:
Tax free allowance designed to cover all the basics (accommodation, food, bills, etc) if you lived in the cheapest area in the country. I don't think it would be 'fair' to make people living in cheap areas pay tax that goes towards giving people in more expensive areas the freedom to keep living there.
After that allowance there'd be a flat level of tax on all forms of income. This would be however much is required in order to pay for whatever government expenditure is deemed necessary by the government.
I think a fair system would be then to get rid of most other taxes, apart from a low level (I don't really know what level) on all business's profits (After all the country has provided police, defense, etc for them, but not hospitals and schools as it does for people). So no VAT or anything like that. Also, some special taxes would be fair - a flat rate road tax to cover the use of roads, and fuel tax because there are good reasons for discouraging people from using lots of fuel.
Hmm, I don't know what to do about gifts... In some ways, money/property that already belongs to someone else has already been taxed, it belongs to someone to do with as they wish, and I don't think it should be the governments responsibility any more. But then in another way it is like an income really - whether people think about it or not, being a grandchild to a wealthy grandad is a sort of service, and gifts might be seen as payment for that service (sure, it might not seem like a service, but then should we not tax people who really love their jobs? That would seem perverse). I think I'll have to go for taxing gifts at the business rate as a sort of profit from the "namehere business".
But yeah, basically a tax free allowance, flat rate above that, a few special taxes, no VAT, and a small tax for businesses.
Or maybe I'm not being radical enough. Maybe a real fair tax would be absolutely minimum, but universal. So police, defence, and anything else absolutely essential to allowing the existence of the country and the rule of the government within it come from a single fee on every person in the UK, from babies to pensioners (children can pay off their tax debt (with interest) at whatever rate they want, but don't have to start until they're adults, get an extension if they are still in education).
Everything else would be charged for or insured for - School fees, fire service insurance, etc.
If you can't pay your bills or taxes, then you could get a government loan, but then if you can't pay it back the government gets to use you as a slave whenever you aren't at your usual job, and pays off your loan at whatever rate they want (you'd have to sign a contract at the start saying what this rate is). If that rate isn't enough to cover your loan repayments then I suppose higher paying options like conscripted slave mercenary, non essential organ donor (would be payed by the health companies), new drug tester, etc would be available.
Maybe living in that system would be a choice - you could live in a completely unregulated zone with no police or army or anything if you wanted - say they just deregulate some large chunk in the middle of England, or they could deport you to any country that would have you, and if nobody wanted you then tough.
I think some services would have to be owned by the government (and rented to companies that would bid on the basis of how much it would charge the users combined with ability to deliver, and with fines if the company didn't keep the service in good repair) or else unfair monopolies would develop, such as roads, water, electric grid, train lines etc, but the users would pay enough to completely support these networks.
The consequences for not being able to pay your subscription to be a UK citizen sound dire, but then in purely material terms it's probably fair.
edit: I'm not sure a democratic government can ever really be fair in a purely material way because they wouldn't get voted it, so you'd have to have some kind of undemocratic government installed. I much prefer the system we have today to the idea directly above btw, it just sounds quite fair in a material way. Maybe in a truly fair system luck would also be accounted for - gains and losses due to luck (be it dice, lottery, genes, the bus breaking down and you missing work and not getting paid, etc) would all be taken/covered by the government. I'm still not really sure what fairness actually entails in my own mind. Equal opportunity and treatment in essence I suppose, but deciding what that entails, and how much personal freedom to throw into the mix is unclear to me.