It absolutely is safe. Naturally stay out of the shady neighbourhoods, but that is valid for Greece as well.

Definitely visit Plovdiv and Veliko Tarnovo as well, as the most interesting cities to visit as a tourist here imo. If you have time, both Burgas and Varna have very large gardens along the sea which are lifesavers during the summer heat. Varna is considered the "seaside capital" and is overall a nice place.

On the seaside, Balchik, Nessebar, and Sozopol are more historic, smaller towns. Coming from Greece, our seaside won't impress you much otherwise ahahha.

In the mountains, definitely visit the Seven Rila lakes and the Rila monastery. A hike anywhere in the Rhodope mountains would give you awesome views and a very pleasant experience of nature, without having to go too high up. If you're in Sofia, hike up the Vitosha mountain, it's super easy to get there from the city and worth the view.

If you visit in August, maybe consider the Open Buzludzha Festival in the Balkan mountains next to the Buzludzha monument. This year it'll be 14-16 August and it's a pretty cool, smaller scale festival, created to support the adaptive reuse of the monument itself (google it to see pictures, it's like a UFO landed on the mountaintop). In the same region (Stara Zagora region) there's also the Rose valley and the Rose festival around the city of Kazanlak, and the schedule should be on their municipality's website. Also there is the Valley of the Thracian Kings with multiple burial sites and tombs throughout, most of them open to visit. In Kazanlak itself there is the Thracian Tomb of Kazanlak that is a UNESCO site and very interesting to see.

Close to Varna, there is the only naturally occurring desert area in Europe. It's filled with stones in funky mushroom shapes and it's an unusual place to visit.

The Belogradchik Fortress to the north west is imo one of the most impressive places in Europe, as it is a fortress site built into the rocks (google it to see a picture, it's worth it). Sometimes they have open air performances there and it's truly magical.

Melnik is the smallest town in Bulgaria, and there you can see our traditional residential architecture. There's also Koprivshtitsa that is a typical town from our Renaissance period (not the Italian one, ours was during the 18. century) with important historical significance. Similarly, there's multiple small villages in the Balkan mountains, such as Bozhentsi or Arbanasi, that showcase our traditional houses.

There's many more places to see and events to take part in, but this is getting very long and I'll stop here. Don't forget to try the food in every region you visit!