on the first �2,020 - 10% (the starting rate)
on the next �29,380 - 22% (the basic rate)
on any income over �31,400 - 40% (the higher rate)
National insurance is a bit more complicated, assuming you are an ordinary employee and not self-employed, look here for the rates: http://www.hmrc.gov.uk/rates/nic.htm
A good rough way of doing this is to assume 30% is taken in PAYE & NI.
If you are paid weekly or multiples of weeks - take your gross pay and deduct �90 (or multiples of �90) and divide what's left by 30% eg �210 less �90 = �120 / 30% = �36 tax and NI
If you are paid monthly take gross pay and deduct �390 and devide what's left by 30% eg �1200 less �390 = �810 / 30% = �243 tax & NI
This doesn't take into account the lower or higher tax band but works with the vast majority as basic rate tax payers.