Gelderland started as the duchy of Gelre, based in Zutphen but named after Geldern in present-day Germany. It was one of the provinces in the Union of Utrecht that declared themselves independent of Spain in 1579, thus starting what would eventually become the Kingdom of the Netherlands.
Dutch is the basic language. In the Achterhoek area, some people speak a local dialect (Achterhoeks), but even they generally speak regular Dutch. German is decently understood and English is spoken by most people.
To get to Ede, Arnhem and Nijmegen, take the Intercity train to Nijmegen which leaves from Amsterdam Central Station at --:22 and --:52 minutes of every hour. Travelling time: 59 minutes to Ede (station Ede-Wageningen), 70 minuten to Arnhem (arrival at --:02 and --:32), 89 minutes to Nijmegen. Note that the train reverses direction in Arnhem; don't be surprised, it will turn south when it leaves the station of Arnhem again.
To get to Apeldoorn, take the Intercity train to Enschede which leaves every half hour at --:27 and --:57, and arrives in Apeldoorn 63 minutes later.
To get to Zutphen or Winterswijk, take the Intercity train to Apeldoorn and change in Apeldoorn to the small diesel train connection Zutphen and Apeldoorn in 20 minutes. It leaves from Apeldoorn at --:10 and --:40, a few minutes after the Intercity from Amsterdam arrived. It usually departs from the opposite side of the platform. Alternatively, you can reach Zutphen via Arnhem by changing there, but it will take about 10 to 15 minutes longer; it doesn't cost more (ie, with a ticket to or from Zutphen you can choose between the Apeldoorn and Arnhem routes). See the article on Zutphen for more extensive information. For Winterswijk, trains leave at --:07 and --:37, and depending on the route taken (Amsterdam-Apeldoorn-Zutphen) or (Amsterdam-Arnhem-Zutphen) will take between 135 and 153 minutes.
To get to Doetinchem, take the Intercity to Nijmegen and change trains in Arnhem. The train to Doetinchem and Winterswijk leaves 2 minutes after the arrival of the Intercity from Amsterdam, at --:04 and --:34, usually from platform 9. These trains take 34 minutes to Doetinchem and 68 minutes to Winterswijk, during the morning and afternoon rush there are 2 more to Doetinchem, at --:15, and --:45. For Winterswijk it's better to travel through Zutphen instead of Arnhem when coming from Amsterdam. To Doetinchem this should take 102 minutes, but it's likely to take 30 minutes longer because there's only a 2 minute change in Arnhem, which one is likely to miss.
To get to Culemborg and Tiel, take the Intercity for Arnhem/Nijmegen at --:22 or --:52, and change in Utrecht for the commuter train to Tiel, which leaves at --:28 and --:58. Travel time is 51 minutes to Culemborg, and 73 minutes to Tiel.
To get from Amsterdam to Harderwijk, take the Intercity for Amersfoort (departure at --:27 and --:57) and change there for the train to Zwolle, departure at --:12 and --:42. Travel time is 72 minutes.
Gelderland is known for the Veluwe, a great place for hiking and biking in all seasons.
In the so-called Bible Belt, certain local delicacies are eaten only by the (elderly) population who also often don't speak standard Dutch. Those delicacies are not recommended to anyone from outside this small culture.
The beer brand "Grolsch" (English: from Grol) founded in Groenlo and based in Enschede in the Achterhoek supplies pilsner in the entire country. Trips to the brewery are interesting, but to drink their beer can be fun as well.
From Arnhem, the German city of Cologne can be reached by high speed train in about 90 minutes.