Originally Posted by
chasm54
The problem is likely to be that you need to adjust the tension on the gear cable. Do this with the barrel adjuster where the cable enters the rear dérailleur. If the lag is when you change up to a harder gear your need to turn the adjuster clockwise - not too much, proceed in quarter turns while turning the cranks and shifting the gears. It's easiest if you have a work stand, but you can do it perfectly well with the bike upside down.
If you are struggling to follow this explanation, simply google for " adjusting a rear dérailleur". There's a ton of videos on YouTube. It couldn't be easier, no need for a mechanic.
I doubt it is a cable adjustment causing this because OP reports it doesn't happen when shifting into a larger cog, only when going into a smaller one, and OP didn't mention continuous chain-skipping during the delay.
More than likely the problem is
drag somewhere in the shifting mechanism. This probably just means that the cables and housing are gummed up and need to be cleaned and lubed. Also it could mean the cable is damaged or there is a kink in the cable where it enters housing and that is causing drag. Another possibility is very dirty cable guides under the BB.
The solution is to clean and lube the cables and housings, and the bb guide, and if they're worn or the cable is damaged, replace. There is probably a youtube video on this.