Begin with soapy water and a cloth - where the black is - see how it goes.
If you need to paint the door use a Satin Finish (Dulux Satinwood is good) - concentrate on just one side of the door at a time - paint the other side another day - remove the handles - gently rub down with sandpaper - just gently mind - clean off all the dust with a brush and vacuum cleaner and then a damp rag - paint the panels first - then the horizontal pieces and lastly the uprights on left and right.
Painting just part of a door is silly; you will never get a match, not even with the same paint - painting over dirt is very silly.
Peccavi, November 2009