Starting Saving: Clear Your Debts & Get an ISA
First of all before you consider saving any spare money you might have you should consider clearing any current debts, especially your credit cards. The simple reason for this is interest charged by loans and credit cards is nearly always more than any you will get back from a savings account.
After you have cleared any current debts then the first place you should look to save is with an ISA. The best thing about these savings accounts is you pay zero tax on any interest made. You are allow a up to £3000 a year in a mini cash ISA. But you must be aware that if you withdraw the money you cannot replace it. For example if you have £9,000 in an ISA but then withdraw then you want to put some back you will only be allowed to put £3000 back in that will get interest off.
TIP: 5th of April is the end of the ISA year so you can deposit £3000 on the 4th of April and get a full years interest. You would then be able to save an additional £3000 on the 6th of April for the following ISA year.