The main promenade along the beach area here at Golden Sands is about a mile or so long and is emmininetly walkable with various bars along the way if your feet get tired! If you don't feel like walking then you can always jump aboard one of the little land trains which cruise up and down the front. The cost, at the time of writing, is 2 Leva but I'm not sure whether that is just for a single trip or will cover you for all day. Me I just trawled up and down betwen bars which was slightly more expensive but more fun!
Try as I might I just cannot seem to find any contact details for transport around Varna which is a shame as the place does have a very good bus service both within town and out to the outlying areas such as Golden Sands. Golden Sands is, however, very well connected to Varna with buses running about every 20 minutes or so from (I think) the Railway Station and (definitely) from the bus stop on the other side of the road from the Dormition Cathedral. The buses to look out for are the numbers 9, 109 and 409 which run from early morning until late and the journey takes about 40 minutes depending on traffic getting out of town.
The bus doesn't actually go into the resort as such but stops just outside, or alternatively at Panorama on top of the hill. The system for using the buses is simply to get on and the conductor will sell you a ticket, which at the time of writing cost 2.30 Leva: just ask for "Golden Sands" if, like myself, you have trouble pronouncing Zlatni Piasatsi!!
Instead of using a taxi, try this unique transportation vehicule. It's a small and very colourful train, which makes it not only useful, but also fun to ride.