Transmission is slow to shift until car "warms up" or has run for about 10 min.

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While accelerating, at the point I'm expecting a shift, it seems to start shifting but it takes a second or two to actually complete the shift. That's what I mean by "slow to shift". The engine continues to rev and I feel like it's not good for the engine. I notice it most going from 1st to 2nd and 2nd to 3rd gear. After the car has been driven for about 5-10 min. it accelerates much easier, without the lag or delay in shifting. I don't notice it being a problem while slowing down. Do I need a new transmission? Or a repair? or to flush the system and see if that helps? Any advice? Please help, Thanks