In simple terms neat cutting oils provide lubrication and not much cooling, whereas soluble oils provide cooling and not much lubrication. Soluble oils are used for general machining whereas neat cutting oils are used for relatively slow but high pressure cutting, such as gear shaping. I only use soluble oils.
I never use coolant for plastics, brass, bronze and cast iron. On the centre lathe I normally cut dry (inserts and HSS) and only use coolant for drilling/reaming with HSS tooling. The exception is parting off – I always use coolant, apart from the aforementioned materials. On the repetition lathe I normally use coolant as tooling is all carbon steel/HSS.
On the vertical mill I never use coolant, apart from WD40 on aluminium as needed. That's partly due to not needing it and partly not having a drip tray underneath, so much of the coolant ends up on the floor. On the horizontal mill I mostly use coolant as I only carbon steel and HSS tooling with heavy cuts by hobby standards. On the CNC mill I mostly use coolant, but primarily to wash away the swarf rather than for cooling.
I always use coolant when cylindrical grinding to cool the work and reduce glazing of the wheel.
Hand and machine tapping is mostly done dry, with some exceptions. I use WD40 on aluminium and coolant on the repetition lathe where I'm tapping at speed – normally 500 to 1000rpm. If the tap is large, >3/4", or the material is difficult (stainless steel) I use Rocol RTD compound. I avoid RTD where possible as it's a pain to clean the work and tap afterwards.
I used to use Biokool14 from Hallett Oils, but now use Hysol XF from Castrol which is a general purpose coolant, formulated for hard water areas. Modern coolants are much better than those of old. I never have a problem with corrosion on the machine tools, but I regularly check the mixture with a refractometer and keep within the recommended dilution. I also never have a problem with nasty nifffs even though the coolant sits in the tanks for months, or years, at a time.
All my machine tools have their own pumps and flood coolant. Dabbing, or dripping, on by hand doesn't achieve anything useful.
Assuming that the OP is in the early stages of learning I'd say forget coolant. On a mini lathe it simply isn't going to achieve much, other than possibly paint a nice streak up the wall behind the lathe. There are many other things to learn that have a far bigger effect on the work produced than using coolant.
Andrew