Do it.
Remove the cotter pin and loosen the axle nut.
Move each adjuster the same amount so the wheel doesn't skew left or right as it puts wear on the chain and sprockets and will affect the handling of the bike. More than 1/8 of an inch and you will feel it in the corners and hear more chain noise.
Simply read your owners manual for instructions.
Leave slack in the chain! Most bikes call for an inch of slack at minimum. Too tight and you will put more pressure on the counter shaft bearing and prematurely stretch your chain.
It really is basic provided you leave slack and adjust the wheel on both sides evenly.
Now on to the next part:
Why is my chain stretching? Did I lube it every two tanks of gas? Have I been dumping the clutch or dragging the rear wheel with the brake?
O ring chains that are well lubricated last longer than chains that are allowed to heat up, for the O rings trap lube between the side plates and rollers, so friction and heat is reduced.
When the chain starts to stretch, it begins to wear out faster, so once you tighten it for the first time, checking it should be included in your pre ride check.
Now is also the time to do a couple more checks.
Pull the chain straight back from the centre of the rear sprocket. A new chain and sprocket set should pull back very little, perhaps an 1/8". Over time when you do this check you will notice that the chain is now pulling even further away from the sprocket which indicates the chain and sprockets are reaching their service limit and you should now read the owners manual and measure your chain to see if it is within spec or needs to be replaced.
Also inspect the teeth of the sprocket for wear. A new sprocket grips the chain perfectly, an older worn sprocket will have worn or deformed teeth, and if it happens on the front and or rear sprocket, it could cause the chain to skip teeth, and potentially lock up the rear wheel which totally sucks in my books.
Chain can stretch unevenly, so adjust your chain, rotate the wheel and check for slack in two other sections of the chain.
So, if you tighten, lube and inspect your own chain, you can stay on top of a good running bike as opposed to putting your safety in the hands of another.
Replacing the chain and sprockets, which should always be done together is a much more involved job that requires research and the correct tools and methods. Dirty but fairly simple when you know what you are doing.
In short, lube and adjust the chain, and check for wear. The more you ride it, the more it will wear.