@Garyt there are several camps on this. I am in the very light grease camp, and my chucks are as tight as the day they were new.
The type of grease is up to you - I prefer a moly or lithium grease, but both of them need refreshing every 7 years or so. A nice red bearing grease will work well, but it has to be a very thin coat.
It is applied with a brush - an old toothbrush or acid brush - very thinly, as in barely enough grease to coat the surface. The more grease you have, the more chips will stick to your scroll and gears - and thus wear out your scroll.
Some really like to use oil. I'm not a fan. It will spray out of the openings in the chuck for a very long time making a line behind your lathe, on your ceiling and in front as well. That being said, chips usually do not stick to the oil film. If you are willing to live with the mess, oil is arguably a little better.
All the best!