I agree with what others have said. You could indeed use Bucharest too.
Air fares over Europe have risen significantly, due to political and environmental factors, and that's why you're seeing such high prices. Gone are the days of €120.
Definitely book the flights first. There's plenty off accommodation available and that's easier to book. I use Booking. com and Airbnb (i also list an Airbnb in Varna).
If you arrive at Sofia, the bus station and rail station are side by side and you can get to them from the airport, either by taxi or Metro.
Trains don't actually go to VT, instead they stop at Gorna Oryavitsa, which is around 6 miles (9.6km) away. So the easiest way is to take a taxi from Gorna to VT. There is a station at VT, but it runs a different route, from Ruse in the North to Stara Zagora in the South. You can change at Gorna and take the train, but there are not many trains, so you could be waiting an age!
Alternatively, you could take the bus from Sofia. Biomet run regular services to Varna, via VT.
Costs... Rail around 18 Leva (€9.23) 1st Class.
Bus is a little more expensive, at around 21 Leva.
If you decide to go via Bucharest, there is an international train that runs into Bulgaria and onwards to Istanbul, that stops at VT, but it only runs on certain days. Otherwise, you'll need to change at Ruse, which would make the journey lengthy.
Another alternative, is to fly to Varna. There is a daily direct flight from London Luton Airport with WizzAir. Then you can either get the bus or train to VT, a slightly shorter journey than Sofia to VT. The Rail Station and Bus Station are in different areas of Varna, accessible from Varna Airport, by taxi (cheap).
Also, check Austrian Airlines, as they sometimes have some good fares, although, you'll need to change at Vienna.
As far as deposits on accommodation go, many do not take deposits, because they prefer cash payment. Hence, Booking. com, cannot handle cash. But be assured, once booked, your reservation is safe - I've used Booking. com hundreds of times without any issues. With Airbnb, they take the payment and then pay the owner direct.
VT has a plethora of hotels, boarding houses and apartments to choose from. I'm not going to recommend any particular one, as it's really a matter of personal taste!
As far as credit cards go, do you have debit cards (bank cards)?? These work equally well as credit cards and I pay with mine on Booking. com or direct at hotels.
There are plenty of ATMs in Bulgarian towns and cities, so accessing cash isn't a problem, but you should check whether your bank charges you to use them.