The rules change if you are due to reach state pension age after April 2010. This is 65 for males, 60 for women, though the retirement age for women increases progressively from this date to move the state pensionable age to women to be 65 also.
Yes, you are correct that you need 30 qualifying years of contributions or credits to qualify for a full Basic State Pension. This is basic state pension we are talking - not Second State Pension which is an add-on.
Yes, if you don't have the necessary 30 qualifying years then you will get a proportion of one-thirtieth for each qualifying year you have.