STOP!
What have you got to start with? You have a lathe, but do you have alive centre?
Do you have a mandrel?
Bushes are metal cylinders with a hole through the centre that thread through the centre on to the mandrel.
Do you have a drill press, if so you may chose not to buy they jaws for your chuck at the moment?
Different pens use different size metal tubes running through the centre, so the drill size will depend on the type of pen you are making. The slimline variety which is the starting point for most, although it can be one of the most challenging (cheapest to cock up though) takes a 7mm drill. The best style of drill is a 7mm dewalt extreme 2 drill bit available from screwfix (they do 7,8,10mm, no imperial in this country unfortunately).
In terms of finish you can pt almost anything on, and you will get some wacky ideas from here if the inmates have been let out on remand. Starting point is a sealer such as chestnut acrylic sealer in a can and a friction polish, again chestnut do a good one.
Think you would be advised to make an investment in a book on the subject, Walter Hall has written a good book on the subject
Pen & Pencil Projects: Amazon.co.uk: Walter Hall: Books, and whilst some of the items are for the future, it'll give you a better overview of your new hobby and perhaps assist with some prudent purchases.