Why when I accelerate my RPM meter fluctuates and the car jumps?

So I bought my car brand new back in 2013 and it has been doing great ever since. The regular checks are done to keep the car in tip top shape. Sooo.... I had my cousin service it September of last year but he changed the engine oil and didn't tighten the bung. He drove it and the engine was destroyed. He got a new engine and had it installed, now recently I have realized that when I move the gear stick to D it delays then the car shakes with noise before selecting. When I accelerate the engine races, the RPM guage goes up, and i feel the car starting to pull because of the power that was sent to the wheels but in an instant the rev goes down and the car still continues to accelerate but each time the rev goes up it comes back down it feels worse so i have to ease up the press the gas again.

I took it to the mechanic and a full diagnosis check was done and nothing showed up. We flushed the transmission and cleaned the oil filter and changed the oil but the problem is still there.