As for coolant, it really depends upon your feed rate of the tool, and RPMs-vs-diameter (feeds and speeds). I do a lot of cutting with a lubricant (cutting oil) applied to the surface of the spinning part. Some people get great success with "foggers" or "misters" for cooling (with less fluids applied and a smaller net mess than "flooding").
And yes, if you have High Speed Steel (HSS) tools, or cemented Carbide, you will need a grinder and/or Belt Sander (and ideally a stone to hone the edges).
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Precision Matthews sells "sets" of cutting tools which are listed with the PM 1130 lathe. Those listed, are properly sized for that lathe (the are known compatible and work well with the lathe).
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The lathe is heavy enough, and has enough Horsepower (1.5 HP/575 pounds) to work with the Carbide insert tools they list. However, for small parts, you are likely to get superior surface finish with properly hand-ground cutters.