I personally hated the rentals when I was learning. But then again, the rentals in Michigan are complete crap. I know some of the bigger resorts rent out actual consumer boards so that would be cool.
When I bought my own gear, it made a world difference. I only ended up using a rental for a day because right off the bat I thought it felt too heavy and stiff. Besides, you'll need to buy a board soon anyway so why throw away your cash towards a rental? Let's see, around here the rentals are $30. 3 sessions is already $90. You can find very good boards from previous years for under $200. With your measurements, you need a board that is longer than 159cm and I suggest you stay under 166cm. Also, your foot size is on the borderline of needing a wide board. I generally recommend people with size 12 and above to use a wide. I am 5'10 180lbs and I use a 159. For you, considering you are a beginner, I would go for a 162-163. You could do a regular board as long as it is not too thin at the waist. I suggest at least a 26cm waist width. Just remember, there are tons of people out there who base board length purely on height alone. That is the wrong way to determine your board length. Weight plays the largest role in your board choice. The next most important factor is your riding style. Shorter for park, mid-size for all-around, and longer for powder. Height plays plays virtually no role in board length. Unless you are 6'11, 100lbs, and with a 40" stance lol.
Some boards to consider (by all means look around for it, I just grabbed the first cheapest one I saw):
FLOW MERC SNOWBOARD 09 [FLOW MERC BOARD 089 BOMBS] - $169.00 : Snow Sports for Less, Quality for Less
K2 Format Wide 2009
2007 Flow Mainframe All-Mountain Snowboard
Plenty more out there
Brands to consider:
GNU, Lib Tech, Burton, Flow, K2, Ride (owned by K2), Forum
BTW, make sure you get yourself a boot that fits snug. Put on some snowboard socks, and try the boots. Balance from your heel to toe for a few minutes. Walking around won't matter since you don't walk while snowboarding. Getting the feel of the heel and toe is most important. They will be pretty snug at first so don't be tempted to go up a size. Now if they are cutting off your circulation, go ahead and move up lol. Just keep in mind the boot will stretch out quite a bit as you break them in. A lot of liners will also mold to your feet when your feet warm them up from riding.
Stick to it, I hated my first day. You'll feel some pain, that's for sure. Good luck and enjoy the ride!