I lived in Maastricht for a year when doing my MA, travelling from various bits of the UK at various times before, during and after. By *far* the easiest way to get there is Eurostar to Brussels, then the hourly IC to Liege-Guillemins and it's usually a 7 minute transfer to the S47 to Maastricht IIRC. Flights to Maastricht airport from London were only on Mondays and Fridays when I went there, most people flew from Eindhoven or Brussels airports instead. On the way to Maastricht I'd recommend stopping in Leuven, although Brussels is also worth a couple of hours for lunch. I personally didn't like Liege, although that was partly because the traffic was bad with the tram construction.
Maastricht city is great, and if you exhaust the bookshops, cafes and churches (the Basilica of St. Servastius is worth going round, and the terrace outside the City brewery has an excellent townscape with tremendous beer) hire a bike and cycle along the Maas for a bit. I'd also agree with the recommendations for Aachen - get the 300 bus across, at off-peak times it takes less than an hour, and you can also hop off in Vaals for a short hike up to the highest point of the Netherlands (it's the tripoint with Belgium and Germany) if you want. Aachen Cathedral and Treasury is excellent value.
Amsterdam would need to be a full day trip to be worth it - the train takes more than 2hr each way IIRC. I wouldn't advise going there if you don't have much time. If you want to do more of the Netherlands (well recommended!) I'd suggest doing that as a separate trip and staying somewhere like Utrecht, which is nice and central and lets you get around to Amsterdam, Rotterdam, The Hague, Nijmegen/Arnhem etc really easily.