You don''t say what you are riding. If it's a bike from the '70s with down tube friction shifters, your mechanic is stretching the truth. If you have a modern bike with index shifting on the handlebars or brake levers, he is completely right. IN the old days of just 5 cogs spaced far apart and shifted by feel with shifters close to the derailleurs and almost no cable housing, shifting was not a precise operation. Using very mediocre cable housing that compressed real amounts was not a big issue. But now we have 9 to 11 closely spaced cogs, teeth and chains enabling the chain to jump onto and off of the cogs easily. And the shifters are twice as far away and have around 2 feet of cable housing with several significant bends. If that cable housing isn't allowing the cable to slide exactly as the shifter intends and the derailleur demands, the chain will not end up exactly over the intended cog. In the old days, that mattered little except making noise. Now you get delayed shifts, unintended shifts and even the chain wandering back and forth between cogs.
So, without seeing you bike, if it is a modern bike with index shifting, that mechanic is probably right on. Now, if our shifting is now perfect, wait a while. But be on the lookout for the misbehavior I mentioned.
You can replace your own cables. Buy good quality housing and cables. (Do this with new cables. You may be cutting the housing longer than the old to get the best big, gentle bends and the old cable will now be too short. Plus the cable is the regular replacement item here, No point of doing all this work and leaving an old one in there.) Tools needed are good cable cutters (bike specific like the Shimano ones are nice but not required), a sharpened spoke, a file and the appropriate hex wrenches. (3, 4 and 5 will probably do you.) Go on-line or to a bike coop to learn how to start the cable at the brake lever.
When you cut the housing, dress the housing ends with the file and stick the sharpened spoke in to bend the metal out of the way of the cable. (That bend can break cable strands quickly. A broken strand or two will really mess up the shifting.) These are all tricks you can see online or learn at the coop.
Ben