The DE series still exists and has a wide range: http://www.terasic.com.tw/cgi-bin/page/archive.pl?Language=E...
The DE0 Nano is pretty cheap: https://www.digikey.co.uk/product-detail/en/terasic-inc/P008... the more capable DE10 Nano looks interesting too: https://www.digikey.co.uk/product-detail/en/terasic-inc/P049... only slightly more expensive.
They provide a good plug and play solution. Program over USB, software available free from Altera (Ages ago Mentor used to do a free version of ModelSim in conjunction with Altera, something similar is probably still around).
You can also just grab Icarus Verilog (http://iverilog.icarus.com/) it's an open source verilog simulator. Will allow you to get to grips with verilog without needing the hardware. The main problem is you may start building designs that simply cannot be built in hardware (plenty of ways to build stupid circuits in verilog).
There are few resources on the internet (comparing to what's available for learning programming). Maybe try 'Digital Design and Computer Architecture' by David Money Harris & Sarah L. Harris.