when and how to change transmission fluid myself?

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My car has 195,500 mi. and has never had a transmission fluid change. A mechanic told me to not change it becuse the milage is high and due to never changing the fluid there is probably a lot of seditary deposits that will be shifted around if he drains the fluid, thus possibly causing future transmission problems. I just had all the belts and water pump changed, new spark plugs and timming belt. Now I feel a slight shift when accelerating that I used to never feel or noticed. When I took my car for an oil change at Jiffy lube, they recommended a trnsmission flush costing $125 dollars. What should I do?