I love Kauai and we visit it often. Every two years we go to hawaii and spend at least a week on Kauai. Last time it was a week on the Big Island and a week on Kauai. Next year it will be two weeks on Kauai. It can be as relaxing or as adventurous as you want. We always spend a few nights on the south shore and then move to the north shore for a change in scenery. That said, I don't spend a lot of time in Lihue. Here is what I tend to do every time we come out.
First things first, buy Kauai Revealed. It's a great start. Then talk to locals.
Napali Catamaran - tour of Napali coast that leaves from north shore.
Blue Hawaiian Helicopter tour - absolutely a must, stunning views
The National Botanical Gardens
Waimea Canyon self guided tour with a stop at JoJo's Anunue (not JoJo's...the other one by Captain Cook) for shave ice afterwards in Waimea. A hike down to Waipoo falls is fun. There are also some hikes along the ridge lines that look out into Napali but I'll let you research those. Some are approved trails and others aren't. I actually just saw the new Hawaii Revealed 10th edition and he marked "airline trail" in Kokee which is locally known as Kalepa Trail. We did this trail and it was awesome. While most trails have you go they woods and then emerge on a vista, this trail has you on views almost the whole time. I will warn you that at one point along the ridge line where the trail is 12" wide, I saw signs of looseness and it concerned me. Therefore I won't officially recommend it even though it's in the book.
Hanging out and snorkeling at Ke'e beach and Tunnels up north. There are several other spots as well, the guidebooks can show you those.
Restaurants-
Donderos at the Hyatt
Tidepools at the Hyatt
Bubba's Burger in Poipu (Hanalei location closed)
Puka Dogs on Poipu (try the lilikoi mustard)
Food trucks in Kapaa
Kickshaw food truck (check website for location)
Bar Acuda in Hanalei
Kauai grill, the restaurant in St Regis in Princeville
We rent a condo so we cook ourselves a bit, too.