Q: Delay in response time?

If I understand you correctly you push the gas pedal and the RPM’s go up but you do not move and then it catches and you take off. If this is what you mean then you have a problem with the transmission either slipping or has a pressure problem internally causing the delayed gear engagement. With the amount of miles you have then I would suspect the transmission has internal seals leaking or the pump pressure is low. Before you look to get any repairs though just make sure the fluid is not low by checking it when the engine is warm. When you have the dipstick out make sure you look at the fluid and smell it. If it smells burnt and looks dark brown to black then you most likely have transmission internal clutch problems. Check the fluid and if you feel you cannot do this then call a qualified mechanic, such as one from YourMechanic, to come and diagnose it for you.