I get asked all the time whether to buy new skis or boots or continue to rent. Largely it depends on your budget and how often you ski. For me as a rule of thumb if you do one week or more a year seriously think about buying some boots. Rental stuff does the job for the odd day but imagine how many people have worn that boot before you, each with a slightly different foot. These days most boots can be custom fitted and they really can feel comfortable!
In terms of skis I think renting for a while is a good idea. That way you get to try lots of different models before splashing out on a pair. Also once you own skis you need to worry about keeping them waxed and the edges sharp. Generally beginners want a softer ski that is a little more forgiving, but you will require a stiffer ski as you get faster and more confident. So if you are progressing quickly you may outgrow the skis you’re on. Renting will allow you to match a ski to your skill level.
You can generally get very good deals on the internet if you live in the United States. If you need some advice on what would best suit you drop me a line.