Thanks. House completed a while back. Builder did all trim and doors in vivid white. Ceilings in ceiling white. Left it to them.

Yes paint white and remove carpet. Paint is cheap as are professional painters. Painters charge what, say $60 per hour so just work out how long it would take you to do it and include setting up, cleaning before and after, any prep work, etc. Then multiply by 60. If existing floorboards under carpet then that would be terrific. Otherwise if no floorboards then it could not be so straightforward. Too difficult to estimate costs really as we don't know how large the spaces are.

I don’t want to paint the beams I want to paint in between them but as my walls are beamed and the ceiling is vaulted I don’t know whether the whole room has to be painted the same colour or I can paint the ceiling one colour and do the walls in another.

If you do not need insulation installed for the room if you install plasterboard, it would be a lot more cost effective to paint the timber ceiling white, it will lighten the ceiling and give an interesting texture. As per the photos that @Julie Herbert has posted. Plaster board will just be a flat service with no interest. Personal preference...