Congrats on your first cruise!
On my last cruise in November, the "Drink of the Day" was $7.50 with the souvenir cup. The refills, or a regular cup is less, but I don't recall the price for certain. Most domestic 16oz beers were $5.75 ie Budweiser products, Coors Light and Miller Lite 4.95. They also have domestic beer 4 to a bucket and it is a little less. I think regular bar drinks were $4.95 if you didn't want anything from the middle or top shelf so to speak. They also add a 15% gratuity to the drink prices.
Zydecocruiser posted a link to the new updated Bar Menu in another thread. Here
is a link.