Most tuning problems are down to the nut, or your stringing technique. When stringing up your guitar you should have a minimum of three turns of string around the tuner before final tensioning. To achieve this, string up the bridge end and pull each string through to its tuner. Measure about 5 cm past the tuner capstan and cut it. Insert the end and tune up as normal. Stretch each string by pulling it up off the fretboard about 5 cm and letting it go back to the fretboard. Don’t let it ‘snap’ back, just pull it up and ease it back until it maintains constant pitch.
To see if your nut is ‘grabbing’ the strings tune up and using your tuner for reference press down on a string behind the nut on the head side. If the string doesn’t return exactly to pitch the nut is not allowing it to return. The same will happen when you play the guitar, particularly if you bend notes. The strings will pull through the nut and not return. If this procedure shows that you have a problem in the first instance try lifting the strings from their slots and rub an ordinary graphite pencil into the nut slots, each in turn. This will cure mild ‘grab’. If this makes a bit of difference but doesn’t cure it completely try rubbing some graphite grease into the slots.
You can make a home brew graphite grease by scraping some pencil lead into a blob of Vaseline and mixing it thoroughly. If this doesn’t do the trick then you will need to consult a guitar tech to have the nut opened out slightly, especially if you have changed string gauges recently. You can do it yourself if you are at all practical. Just use a piece of very fine wet & dry paper under the string and open out the slot. Don’t go mad. The amount to make a difference is very small. If the problem has been eliminated it is good practice to still lubricate the nut with graphite grease or light oil periodically. Don’t splash it about and keep it off the fretboard. If you do get some on just wipe it off- it’s messy, not life threatening for you or the fretboard.
I often see people complaining about slipping tuners but that is very rarely the case. The way guitar tuners are designed with a worm and cog make it almost impossible for them to slip. If the screw attaching the cog to the capstan works loose it can affect the tuning so just tighten it up again. Sealed die cast tuners as fitted to most electric guitars pretty much look after themselves and rarely fail. Just check from time to time that the screw holding the tuning button on is nice and tight.