Most people fly to Toulouse or Barcelona, which are both well served by UK airports. You can then take a hire-car or a coach transfer (included in some holiday packages) in 3-4 hours. If you have time, a week in Andorra would work well with a short break in Barcelona.
There are no trains in Andorra. If you choose to drive from the UK, count on a drive of about 12 hours from the French ferry port/Eurostar terminus of Calais. Alternatively, take a ferry to Santander in Spain and drive about seven hours.