As stated, at the first timer level (and even Silver to an extent) a lot of things will be fully booked up already. Especially the alcohol tastings - those go fast! There are a couple of things you can do though.
First, be flexible and persistent! Grab what you can on your schedule and then check back every couple of days (if you want). People can and do change their minds and cancel excursions, onboard activities, even entire cruises. So things will open up. This can happen up to a week before you sail! So while you should set realistic expectations, keep your eyes open too!
Next, talk to shipboard guest services when you get on board. For everything except excursions, talk to GS. I've been able to get into "full" alcohol tastings or find out about additional offerings that weren't on the website and get in to those just by asking. For excursions, check the Port Adventures desk. Last minute cancelations might be your best friend! Note that once aboard you cannot cancel any of your excursions, but if you have an open day, you might be able to score something cool!
Finally, check outside of
DCL. A LOT of people "freestyle" it using Viator or other such sites to book non-DCL excursions. You can save a lot of money sometimes that way too. A couple of words of warning there: A) on a DCL excursion, the boat will always wait for you. On a freestyle one, they won't. B) Check the reviews for reputable excursion operators, not just 2 guys and a truck. You want to make sure it's safe and well run. Most of them will be - they need the good reviews to stay in business.
Best of luck to you and hope you have a great time!