Which engines do airliners start first?
For some general reasons, see Why is engine No.2 started first, instead of No.1?

Airbus A350

Specifically to the Airbus A350, I could not find any reason why one engine must be started first. As far as I know every modern airliner could start engines in arbitrary order.

The Airbus A350 has two hydraulic systems. From the Airbus A350-900 Flight Deck and Systems Briefing:


  The A350 hydraulic system has two independent
  hydraulic circuits:
  
  
  The YELLOW hydraulic circuit
  The GREEN hydraulic circuit
  


Each system can be pressurized with an engine driven pump from either engine, or electrically on the ground:


  Engine Driven Pumps (EDPs)
  
  Four EDPs pressurize the hydraulic system. On each
  engine there are two EDPs, one pressurizes the
  GREEN hydraulic circuit and the other pressurizes the
  YELLOW hydraulic circuit. Thus if an engine fails, the
  remaining engine can still pressurize both hydraulic
  circuits.
  
  Electric Motor Pump (EMP)
  
  One EMP per hydraulic system can provide hydraulic
  pressure on ground only, when all engines are shut
  down. For example, the EMP of the YELLOW circuit
  operates automatically for cargo door actuation.


The main wheel brakes are powered by both hydraulic systems:



And also the parking brake uses pressure from both accumulators:



Which engine is started first has therefore no impact on which hydraulic system is powered first or which type of braking is available.

Airbus A320

The situation is a bit different for the Airbus A320. As Jan pointed out in the comments, it is normal to start the right engine first, because it provides hydraulic pressure to the yellow system. From the A320 FCOM Hydraulic chapter:


  Green System Pump
  
  A pump driven by engine 1 pressurizes the green system.
  
  Blue System Pumps
  
  An electric pump pressurizes the blue system. A pump driven by a ram air turbine (RAT) pressurizes this system in an emergency.
  
  Yellow System Pumps
  
  A pump driven by engine 2 pressurizes the yellow system. An electric pump can also pressurize the yellow system, which allows yellow hydraulics to be used on the ground when the engines are stopped. Crew member can also use a hand pump to pressurize the yellow system in order to operate the cargo doors when no electrical power is available.


The yellow system powers the alternate/parking brake system:



However, starting the left engine first is still perfectly possible. Hydraulic pressure from the accumulator is still available after pressurizing the yellow system with the electric pump before pushback to provide braking.

Boeing 737

The Boeing 737 has two hydraulic systems: A and B. Both can be powered by electric pumps. From the FCOMv2 13.20.2 Hydraulics - System Description:


  Both A and B hydraulic systems have an engine–driven pump and an AC electric
  motor–driven pump. The system A engine–driven pump is powered by the No. 1
  engine and the system B engine–driven pump is powered by the No. 2 engine. An
  engine–driven hydraulic pump supplies approximately 4 times the fluid volume of
  the related electric motor–driven hydraulic pump.




The braking is provided by both systems. From the FCOMv2 14.20.3 Landing Gear - System Description:


  Normal Brake System
  
  The normal brake system is powered by hydraulic system B.
  
  Alternate Brake System
  
  The alternate brake system is powered by hydraulic system A. If hydraulic system
  B is low or fails, hydraulic system A automatically supplies pressure to the
  alternate brake system.
  
  Brake Accumulator
  
  The brake accumulator is pressurized by hydraulic system B. If both normal and
  alternate brake system pressure is lost, trapped hydraulic pressure in the brake
  accumulator can still provide several braking applications or parking brake
  application.


Again, there is no reason to start one engine before the other because of hydraulics/braking.

Conclusion

Engine start order is mostly tradition and will be specified by the airline op-specs. The shutdown oder also does not matter since both engines are usually shut down at (almost) the same time at the gate, except for a single engine taxi where the hydraulic system that powers the nose-wheel steering must be pressurized.