Lara, have you registered with a GP? That puts you onto the NHS system, not the National Insurance number. The NI is for your tax code. Here's the link on how to register with a GP
http://www.nhs.uk/chq/Pages/1095.aspx?CategoryID=68&SubCategoryID=158As for dentists, you'll need to check your area if there's an NHS dentist who can take you on straight away- If you're needing:
•veneers and palatal veneers: new surfaces for the front or back of a tooth
•inlays, pinlays and onlays: used to restore damaged teeth
It looks like it costs £204 in England on the NHS.
Otherwise, if you can't get an NHS dentist, then you'll need to go private. My private dentist is actually very much the same pricing as the NHS. Likewise, Andee's suggestion of a dental school is a great one.