Move data from NIOS to hardware and then back to NIOS

No, if you use a DMA you don't need PIOs or dual part memories. One way to do it would be to use an Avalon Stream interface between the DMA and your hardware. You may need a FIFO between the two in some cases but this can be added in SOPC builder.

The PIOs are fine for slow systems with little amount of data but they are less efficient than a DMA.

Using a dual port memory and an Avalon slave interface should be easier to make, but it seems that you will handle a lot of data so it may be impossible to use on-chip memory to store your data. That's why I'd recommend a DMA.

If you are new to Verilog you may want to start with something simpler than this project though.