I find that it depends on the plastic, much like wood. Some plastics are relatively soft and don't chip very much, others are very brittle. You'll soon learn which is which.
The lathe speed is less important than the sharpness of your tools and the agressiveness of your cuts. I used lower speeds when I started turning plastic pens, lately I've been using higher speeds.
When you start rounding the blank you need a very light touch, you'll get small chips flying everywhere. As it gets rounder the chips will be less, the sound will change (chipping sounds like scratching or chipping glass) and you'll get more ribbons of plastic.
Once round I use a skew to get near the final shape and then a scraper to do very fine final cuts to shape. (I need more practice with a skew:redface

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I then wet sand starting with 400 wet & dry, 800, then all 9 MM grades. I cross sand on the 400, 800, and every 3rd MM. The wet sanding will stop any overheating. Finish with Brasso and liquid car polish (non-abrasive liquid wax).
When drilling, squirt water into the hole to cool the bit and the plastic. Drill 1/8" - 1/4" at a time and clear the bit. Sometimes the water and swarf become paste like and sticky, especially near the end of the drilling, so be aware of that.
You will also need to check the transperancy of the plastic. The blank may look opaque but when it is only 0.5 - 1mm thick you may see the tube and glue. If unsure try cutting a thin piece off the end, sand it smooth (400 w&d) and hold it to the light. If transparent you'll need to paint inside the hole and possibly colour the glue as well.
I only started painting the holes recently and found auto paint works well. I let it dry for an hour or so, then do a second coat and let dry overnight. I also use the auto paint to colour the thick CA for glueing the tubes. 3-4 drops of paint in a pool of thick CA on a plastic bag, mix and use. The auto paint doesn't set off the CA. I use earbuds for painting the holes, and the plastic stick from the earbuds for mixing and glueing. You may also need to drill your holes fractionally bigger to account for the thickness of the paint (I use a 7mm brad point that is slightly under, followed by a 7mm jobber that is slightly over size) or sand the hole a bit.
Some of this may be a bit obsessive but ......

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