French press is an immersion brewing method, also typically unfiltered. It produces a mellower, more full-bodied brew. Pour-over is a percolation brewing method, typically filtered. It produces stronger, clearer flavors with less body. Both are excellent, cheap alternatives to using an auto drip machine and will absolutely produce better coffee than most pod machines. Which brew method is better depends on personal preferences.
In my experience, pod machine coffee (specifically Keurig) tastes way worse than auto drip machine coffee, which is worse than the best coffee that you can make via manual pour over.