You can indeed use train to visit nearby countries, most of which you do not need a passport to enter - i.e. no border checks - but as you should have ID with you anyway. Distances are beyond what a strippenkaart allows so you must buy seperate tickets. Do bear in mind that the journies will take several hours each way so don't be too ambitious or all you will see is a view from a train. You can get more info on international trains from the dutch railways website
www.ns.nl, look for the english option at the top of the screen.
For the UK you will need a passport and possibly a visa depending on where you are from. As ABC123 says London can be done in a day - I regularly do so for business - but you will be terribly rushed to see even a fraction of what is there. From Schiphol you can fly to 5 airports surrounding London all with public transport links to the centre:
- City Airport is closest to the centre but flights are most expensive, look at VLM or KLM airlines
- Heathrow is the main international centre but it's very congested and the train in is expensive
- Gatwick, Luton and Stanstead are the other options: EasyJet flies to all from AMS
Flying mid-day will get you the best rates.
My own preference is usually Gatwick where there is a station attached to the terminal and a choice of stations in London: Waterloo, Victoria or Blackfriars depending on where you are going.
Be aware that the UK is expensive and London especially so. You will also need to change money to sterling whereas other countries surrounding NL also use Euros.