By air:

Nearest airports are Milan's Malpensa airport (http://www.sea-aeroportimilano.it/en/...) and Turin airport (http://www.aeroportoditorino.it/EN/vo...).  To reach Biella from Malpensa airport using public transport, take a bus to Novara (schedule on the airport website), and a train from there (see below).  From Turin airport, take the SADEM bus to Turin's Porta Susa station (http://www.sadem.it/upload/orari/esti...), hence a train (see below).

By train:

Biella has a train station (Biella S. Paolo), with frequent service from Milan (via Novara) and Turin (via Santhia') -- http://www.trenitalia.it/en/index.html

By bus:

There is a good, direct bus connection to and from Milan's Molino Dorino subway stop (2 buses a day Mon./Fri., one Sat.) -- http://www.atapspa.it/orari/go/cerca1...

By road:

 There are excellent road connections, with the nearest autostrada exits at  Carisio, Balocco, or Santhia', on the A4 (Milan/Turin).