Faster 2005 Corolla Xrs

You might want to give a ballpark budget. There's a wide spectrum of options in an even wider spectrum of costs...

Assuming you are focused on acceleration (braking/handling not included):

The 2ZZ engine itself is pretty much optimized, but there are minimal gains attainable by increasing intake and exhaust flow. Forced induction provides big gains, but costs significantly more.

Cheap/near free options:

• Make the car lighter. Remove unnecessary weight and/or begin removing unneeded interior
  
• In warmer climates, throttle body coolant bypasses are an option. Not much to be gained, but it is easy to do and costs practically nothing.
  

Works on any car:

• Lighter wheels and tires. If you have to spin up four giant flywheels to accelerate your car, might as well make them lighter.
  

Improving normally aspirated performance:

• Header ~$300+
  
• Higher flow exhaust (likely that performance gain is more from having a lighter muffler than better exhaust flow)
  
• Larger throttle body ~300+
  
• Improved tuning (piggyback or standalone ECU replacement) ~$500-3000 for hardware, then add tuning cost.
  Lotus manages about 189HP from the same engine with a more aggressive tune and simple headers (not sure if it is the same throttle body).
  

Forced Induction ($):

• The 2ZZ internals are strong enough to handle moderate boosting. Lotus uses same internals on the supercharged Elise/Exige variants - up to 265 HP.
  
• I have a friend that ran a 2ZZ with stock internals (only added stiffer valve springs) over 400HP:blink:
  
• Cost depends on build. Understand that "kits" that do not include injectors, (likely)fuel pump, and ECU remapping or replacement will require that at extra cost.
  
• Lotus Elise/Exige owners can buy supercharger kits from ~$4000(~250HP with ECU reflash) - $7000(~300HP with standalone ECU). Giant shoehorns to make it fit in an XRs not included (would require custom brackets, etc).