Think of the paint or varnish as having 2 functions. One is to protect the wood and the second is to provide a decorative finish.
If the new door has already been varnished then you dont have to do anything if you dont wish. If its bare wood then it must be painted or varnished immediately it is hung and before it gets wet.
If you like a wood finish then use varnish, if you prefer colour then use a paint.
To varnish just decide if you would like to stain it a different shade first (there is a large range available). One coat of stain then 2 coats of a good exterior varnish will do. Gloss varnish if you like it shiney, Satin varnish if you like it less shiney.
Most woods darken a few shades anyway when you apply varnish so its likely you wont want to stain the door but just apply a few coats of varnish. Apply a coat to the never seen bottom of the door, wait an hour and see if you like the shade it turns. If not then choose a stain and repeat the test.
If you prefer colour then simply apply an undercoat first then two coats of an exterior wood paint of your colour choice. Once again its gloss if you like it shiney, satin if you want it less so.
A 2 inch brush should be fine for the job, also use a half inch if there are any fiddly areas such as brass door plates or key plate.
Personally if it was me with a brand new wood door I would just give 2 coats of a decent satin varnish using a two inch brush, but front doors are funny things and people do the oddest things to them.
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