There is no really easy way to get data out of an FPGA to a computer. The signal tap works well for one time use. There is also a memory editor that works like signal tap, that allows you to read/write to memory in the FPGA thru JTAG. You have to select this option in the megafunction for the memory. The function is available under TOOLS.
When talking to the computer, my favorite way is the old serial port. Two pins on the FPGA with an RS-232 driver gives you easy access to the FPGA. You can use a simple terminal emulator to check that the port is working. I like termite (terminal emulator). The code for the UART is about a page each for tx/rx. OPENCORES has code for a UART, but they are usually fancier than you need. Also, MATLAB can talk directly to the port.
Second on the list would be USB. FTDI makes a parallel/serial to USB converter IC. They also provide the drivers for windows, so it looks like a serial port. You have to have a way to hook this up to your board, which may not be easy. Mouser sells demo boards and the chips. Again, you can use a terminal emulator to verify that it is working.
Third on my list would be Ethernet. If you are using the Stratix III demo board, there is an Ethernet port. If you look in the demo files for the board, I believe that there is a demo design on the CD for the ethernet port. I not a software person, nor do I much experience with NIOS, but I imagine that there is a learning curve with any choice you make.