The easiest way to use your phone in Bulgaria is to purchase a prepaid Sim Card. It costs only 6 leva for 1 GB of data or 8 leva for 2 GB.
I bought a Sim cards for Liat and me at the Telenor mobile network shop in the City Center. It took the seller approximately 5 minutes to make some setting definitions so it would work, and that's it. Definitely a successful deal.