What exactly should I be looking for? Something like this?
Yes. But be careful of the shank size when you buy, and check if you have the holder first
If you didn't bought anything else while I was not looking, you have a 1/4" square holder for the lathe. And 3/8" round for your mill (3/8" end mill holder and 3/8" max on the drill chuck).
You link show a 1/4" round
shank. To my knowledge, Sherline only makes a 3/8" round Tool post
for the lathe. I don't know why, though make a 3/8" to 1/4" bushing adapter is not a big deal.
So your only alternative for using a 1/4" round chuck would be to use it on the drill chuck + mill, with your piece to groove on the rotary table. You could have used collets
as a very good way to hold 1/4" chank, but for cost reasons they are not in your basic set
Or you can use another end mill holder
size. Additional costs again.
As a rule, try to avoid cutting anything that will make a strong lateral force on the drill chuck, like with a end mill for example. Drill chucks are made to take a longitudinal force, not a lateral one. You will ruin your chuck precision if you do it. You can do it while you have not other accessories, but with very light