My advice to you Bill would be to never use mild steel as I never do. I am sure that those reading this and know the type of things I make will wonder just how do I manage. Let me tell of a situation I had early in my metal working activity, that being when I became editor of MEW.
I was making a device that needed two quite complex spindles that ran in cast iron bearings. I made the first and was very pleased with the result. However, when making the second I could get nowhere near the finish I wanted for the bearing part of the spindle. Why was this.
Well, the first was the last piece from a bar whilst the second was off a new bar, obviously two different grades of steel. To make the situation worse I did not know what either of the two bars were. From then on I would not purchase from a supplier who just stated "Mild steel" or even "Free cutting mild steel" I would go elsewhere.
As to your question regarding grades of steel to be used in the home workshop, this is difficult to answer precisely. Typically, your tapers are unlikely to get anywhere near the amount of use as would be in industry so just a common workshop steel will be OK. If though you were making a centre then a steel that could be hardened should be used. Having said that, unless you have the necessary equipment for making such an item, purchasing the finished item is all that can be done.
Having said I never use mild steel I will come clean and say that after my experience I never purchased steel without being told its type. Any that I had in stock I would mark the ends G good. P poor and VP very poor as I used them. I do of course mark the end of any steel I now purchase.
With regard to material types these are the common general purpose types. That is, not for hardening or for strength
230M07 (EN1A) Free cutting mild steel, available in round and square bar , also in flats if you search around
230M07pb (EN1Apb) Leaded, very free cutting steel, available in round and square bar.
070M20 (EN3B) Turns and mills but not as easily as the two above.
Those in brackets are the earlier numbers
I am sure someone will expand on these material types.
Harold