Of course your host family provides you two complete set of bedding and towels. Speaking about your everyday needs, we advise you to bring only the medicines you use if any. The other things might be easily found and bought here.
St. Petersburg now is different from St. Petersburg of about 10 years ago and the choice of goods is practically the same your have in your own country. We also advise you to bring some small souvenirs traditional for your country as the possible presents for the host-family or teachers.