I have a new 05' Accord 2.4 liter. What oil do these engines like best? Synthetic or Dyno, or both? Which brands and weight?

5W20 is what Honda says to use. Dino or synthetic is your call. My personal choice is Mobil 1 synthetic but all oils made today are light years ahead of those made just five years ago. Two very good choices in dino oils are Motorcraft and Havoline but just about any API rated oil will do quite well. Filters DO matter and Fram is among the most poorly constructed.

5w20