I can't disagree with any of that
However, you'll probably also need some cymbals, unless the second hand kit includes them; they're not cheap and at a minimum you'll need a hi-hat pair and 1 crash/ride. Again, try second hand. Good brands are Zildjian, Sabian, Meinl, Paiste and Istanbul. Pearl used to do a budget range but I don't know if they still do
Like with most things, buy cheap = buy twice, but especially cymbals.
Finally, remember that a drum kit is not a quiet – or small – instrument. You can get practice pads to help muffle the sound but if practicing at home you'll need to consider the neighbours. A full size kit can also take up 6ft x 6ft of floor space or more.
If noise is a consideration then it may also be worth trying a s/h electronic kit from Alesis, Roland or Yamaha. Again, not cheap.
Have a look at http://www.drumwright.co.uk to get an idea of prices