The car is a Dodge Neon SRT-4 with about 81,000 miles that is used for amateur occasional drag.
What grade and type of oil would be the best to improve my performance while still preserving the engine for daily drive during the summer period?
If you are looking to improve your cars performance, then I suggest you try something different. Your gains from using a different viscosity oil will be laughable (if there will be any), especially compared to any proper performance-oriented change. Unless, of course, if what you can do with the engine is limited by regulations and you are exploring the last remaining possibilities to get those extra split seconds.
You should always use engine oil with the viscosity specified by your car's manufacturer (unless you have been doing something with the engine and/or oil pump), definitely NOT with higher viscosity.
Using synthetic oil instead of semi-synthetic might be a good idea, if you want to prolong your engine's life. It usually has better properties and retains them for longer.
It probably should also be noted that semi-synthetic oil actually is mineral oil mixed with synthetic in unspecified proportions. So you really never know what you are paying for unless you really trust the manufacturer for some reason, or you have a mobile lab handy.
I know some Honda guys that run 0W-20, but that is much too thin for the 2.4L SRT motor. People in the Neon community run 5w-30 or 10w-30 (pretty much the standard for turbo 4bangers nowadays).
There is risk of cars that have been running non-synthetic leaking through seals after switching to synthetic oils with higher detergent content. However, I have done this on many 100k+ mile cars and have never witnessed this. Turbocharged cars should always run synthetic, as standard oil has the tendency to coke up the oil lines near the exhaust end of the turbo.
Personally I would make the switch to full synthetic - Royal Purple or Amsoil. If you burn a little, no big deal - add some more.