It is a superb scenic town with cobbled streets, museums, ruins and ancient churches. If you do make the effort I suggest that you go early as it gets very busy in the summer.
Burgas is a busy economic town similar in size to Varna but anywhere near as attractive. It has a few museums and art galleries; the most worthwhile sight is the maritime park. This is a stretch of greenery that runs along the coast within easy walking distance of the city centre. It has many manicured flowerbeds, fountains, war memorials, sculptures and bars.
The city is surrounded by four lakes collectively known as the Burgas Lakes, these make up the largest wetland system in Bulgaria and is home to about two thirds of the country's bird species. Here you can walk trails, take boat rides, it's a twitchers delight!
Beyond Burgas you can visit Sozopol, another historic and beautiful town with two superb beaches, churches and museums.
Primorsko, 52kms south of Burgas is another busy resort mainly populated by Bulgarians. There are plenty of bars and restaurants and a small beach.
Take the main highway from Varna and head inland, the land is quite barren and uninteresting, until you reach Shumen. This is a pleasant city situated at the base of the Stara Planina ranges. It is an industrial and military city and feels like communism is still at large.
The nearby Shumen Fortress and the massive Creators of the Bulgarian State monument are worth a visit.
From Shumen you can head south into the hills which is a peaceful place to lose yourself in. Alternatively head to Madara to see the Horseman; carved into the cliff in the 8th century is a mounted figure spearing a lion. Another Unesco World Heritage site. Near to the horseman is the 12th century Madera Fortress, if you not into history it's still worth the climb up to get the sweeping views of the surrounding area.