At what point during flight would Concorde go supersonic? Would it climb to cruising altitude first? Does altitude affect the transsonic difficulties (buffet, need for extra thrust, etc.)?
Concorde went supersonic during its climb phase to the cruising altitude.
The altitude at which the Concorde can go supersonic was limited by the regulations. For example, the aircraft was only allowed to go at 400 knots during the climb phase between 6,000 and 32,000 ft; also, the aircraft was allowed to go supersonic only after it has crossed the coastline.
As the air gets thinner during climb, the Mach number increases for same IAS, reaching around 0.93 at 25,000 ft and 0.95 at 28,000 ft. After this, the afterburners are engaged and autopilot is set to the cruising altitude (~ 60,000 ft). During this climb phase to the cruising altitude, the aircraft goes supersonic, settling at around Mach 2 at over 50,000 ft.
The flight envelope limitations of Concorde (at one configuration) is shown in this image:
The altitude affected the thrust requirements, as more thrust is required for the same speed at lower altitudes due to density. This is one of the reasons for Concorde flying at altitude far above that of subsonic airliners.