170-175cm is probably good for your size and weight and skill level.
Seeing as you struggle on greens, I would say ski some rentals until you're at least comfortable with all greens and OK on most blues. Otherwise if you buy a whole set of beginner boots and skis, they will be limited in their applications. What if you decide to quit (please don't. Skiing rocks!), that would be a huge investment lost. Once you progress to intermediate or beyond, those initial skis and boots will likely hinder your performance and you will need new gear. In my experience, the progression from beginner to intermediate is the fastest. The progression from intermediate to advanced and beyond is slower. I'm saying this to illustrate the point that once you get better, you will get more out of a set of skis over a wider skill level range (some skis cover intermediate to advanced skill, but beginner skis never cover past intermediate skill level).
Yes, having your own gear is awesome and much better than rental quality crap, but if I were you I would devote that money to lessons and more time on the hill. Once you get to intermediate level, then consider buying skiis.
If you're loaded, then by all means go for it haha. Welcome to newschoolers and good luck skiing this year man!