My 308 is slow to accelerate - 2008 Peugeot 308 1.6

its in learn mode ,give it time
if not that ,the check all fuses then scan it..
scan the OBD2 port.

https://www.lacrossetechnology.com/media/catalog/product/3/0/308-146_website_manual_121714.pdf


https://www.lacrossetechnology.com/media/catalog/product/3/0/308-146-308-146w__faqs.pdf


https://www.lacrossetechnology.com/media/catalog/product/3/0/308-146_website_manual_121714.pdf

When VDC light is on, it means Vehicle Dynamic Control is engaged and it puts resistant on wheels to reduce slipping.
This is the reason to cause slow acceleration. If you have OBD reader, you may find out why VDC is on; otherwise, ask someone has experience on mechanic for consulting.
VDC issue needs to be resolved or you will consume more gas and wear the car faster.

I have seen many time that the rug/mat may be preventing the accelerator to be completely depressed. Check that...

Suggest oil pump is inoperative?

car keeps on slowing down, when iam driving and changing gears.

use a couple of cans of injector cleaner as faulty injectors are indicated

Click the link below to troubleshoot this problem: ---
Car loses power while acceleration? http://schematicsdiagram.blogspot.com/2011/12/car-loses-power-while-acceleration.html

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Loss of acceleration?

http://technoanswers.blogspot.in/2012/02/loss-of-acceleration.html

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Acceleration lag problem?

http://repairhelpcenter.blogspot.com/2012/01/acceleration-lag-problem.html

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How exactly the Vehicles Accelerator Cable works?

http://repairhelpcenter.blogspot.com/2011/11/how-exactly-vehicles-accelerator-cable.html

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How exactly Vehicles Accelerator Pump Diaphragm Works?

http://repairhelpcenter.blogspot.com/2011/11/how-exactly-vehicles-accelerator-pump.html

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accelerator pedal sensor?

http://technoanswers.blogspot.in/2012/02/2009-chevy-impala-accelerator-pedal.html

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stalls and bucks on Acceleration?

http://technoanswers.blogspot.in/2012/02/nissan-pathfinder-stalls-and-bucks-on.html

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Engine accelerates when brake is applied?

http://technoanswers.blogspot.in/2012/02/engine-accelerates-when-brake-is.html

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This will help.

Thanks.

MISFIRE COULD BE STOPPED UP FUEL FILTER.SLOW ACCELERATE PROBLEM COULD BE TURBO CHARGER PROBLEMS.

I have the same car. So far no problems with the Transmision. I had the Trans oil change out every 60 K Sure sounds like you need to get the Transmision checked. Is the fault light on. I have invested ina fault code reader sure is worth it. 
Good luck.  Vilmos