+1 on the harbor freight set , still using mine years after I bought it especially the 3/4" spindle roughing gouge , 1/4" parting tool and the 1" skew. I have a number of Benjamin's Best and maybe HF tools have changed some since I got mine but can't agree that Benjamin's Best from PSI are any better . My first bowl gouges were from PSI and the hardening on them was inconsistent as were their spindle gouges, which meant sharpening them down a little further until I hit hard metal again. I actually still have them and use them for the bottom of bowls even though the 1/2" is getting short. I up graded to P&N bowl gouges and a spindle gouge from Lee Valley. The 1/2" P&N bowl gouge from Lee Valley runs $39.50 + shipping unhandled . Since you are just starting out the Benjamin's Best is still a good value especially when first learning to sharpen your tools and sharpening gouges will take a little practice. The Harbor Freight lathe you mentioned is the one I started with and is not a bad starter lathe. Be sure to weight it down well and even though the head rotates don't go wild with large out of balance blanks on it.
I know the voices in my head aren't real but boy do they come up with some good ideas !
People discuss my art and pretend to understand as if it were necessary to understand, when it's simply necessary to love. - Claude Monet