Hello, Sudha, and welcome to cruise-chat!
Your first cruise--how exciting. Taking your questions in order; the difference in price between an inside and a ocean view cabin on a 7-9 day cruise really isn't all that much and most definitely worth it. I would encourage you to consider also a balcony cabin; again, not a lot more than an ocean view on a shorter cruise, but being able to sit on your private balcony and have that fresh sea air circulating in your cabin is priceless. Next, a few ports or a lot of ports is completely a matter of personal preference. I enjoy a lot of long, leisurely days at sea. It sounds like you and your wife are more active and like water sports, so more ports are probably best for you. East, West or Southern Caribbean shouldn't matter; the weather should be nice and the water warm everywhere.
Finally, I would encourage you to act quickly; holiday cruises are very popular. The good news is there are many ships sailing this area at this time of year. Princess alone has no less than five sailings on the dates you specify. So book right away and have just a terrific trip.