We make a product for it and I try to stay away from anything that could be seen as marketing or sales. I’ll PM you the ones we make.
In general though you want a low viscosity, non-tacky oil. Usually penetrating oils work well for this. The idea is to clean all the tapers and leave behind a layer of that oil as it will help protect against oxidation (increase friction and wear) and provide a bit of lubrication. If you can find one with dry lube in them (PTFE, boron nitride, tungsten disulfide, etc.) those will work better long term.
Process wise you want to apply and then remove all but the mono layer. So wet applicator with oil rub into everything. Get a dry applicator and remove all excess.
For to the inside tapers (spindle and tool holder) it’s best if you can spin them up and use a cotton swab to apply and drag from top to bottom to remove debris. Probably can’t do that with the spindle though as most ATC ones can’t be run without a tool holder. I’d also use a cotton swab for the inside of the nut taper and threads. For the outside tapers something like a microfiber cloth to apply and then remove all the excess. This would be for the collet (both ends), and toolholder.
Variations on the above would be if you have a build up of lacquer, old oil, or dirt you want to repeat this until the cloth/cotton swabs are clean when removing the excess. If you have a dry lube in the oil multiple applications to get it to fill the gaps.
Do NOT get any in the inside of the collet bore. If you do use something like acetone or brake clean on a cotton swab to remove the oil. Yes it will make that surface prone to oxidation and all of the above. But it’s the weak point of the slip resistance and you need it free of lubricant.