Been to Kauai over 10 times you are going to love it. Must do's would be visiting the beaches on the North Shore and Waimea Canyon (go up to Koke'e State Park first to visit the Kalalau Lookout as it gets foggy sometimes later in the morning and then visit the Canyon on your way down). Lots of great hiking up in both those areas.
In the summer, look into staying on the North Shore, that is where the water will be calmest that time of year. I know a lady who owns a condo at Hanalei Bay Resort in Princeville and it is a gorgeous place to stay. Beautiful pools with a waterfall, nice BBQ areas. All the places in Princeville are up on a bluff but you will have a nice ocean view and also maybe a view of Bali Hai. You can walk down the path to the beach, or they will also take you down or bring you back up on a golf cart. Rent from an owner only, the hotel and timeshare rooms need a remodel (scheduled for later this year).
Boats should be going out of the North Shore that time of year, plenty of good companies, no personal experience but Na Pali Catamaran gets good reviews and can go into the sea caves (conditions permitting). Have been out on Captain Andy's and also Holo Holo but they leave from the west.
For car rental try this site, last trip I got a nice new intermediate from Avis:
www.discounthawaiicarrentals.com
Enjoy and have a wonderful honeymoon!

We love the North Shore (Bali Hai view) and spend the 2nd week of our Honeymoon there. Check VRBO for some great homes/cottages on the beach along Hanalei Bay and out toward Ke'e beach - or the various condo/timeshares (also - see Cliffs Club) or the St. Regis for upscale/great location in Princeville.

All thumbs on a laptop at the moment or would write more.

I don't know if it gets any better than this - http://jalbum.net/en/browse/user/album/926701;jsessionid=pz0toczrchvc1k922gx1plfj6

I would look at renting a house or cottage in Hanalei. It's beautiful and the beach is great. Plus, you could walk to restaurants and/or bars.

Aloha from Kauai...since June is stellar on Kauai's north shore, i'd say a cozy little cottage anywhere from Kilauea to Haena. Plenty of private accommodations away from resorts/Princeville condos/timeshare, that are affordable, awesome views,within steps of the beach and private/clean/fully stocked. Places i've booked for guests have bikes/surfboards/SUP/snorkeling gear included... Be dollar savvy and save your money for sushi at the Dolphin(go see Jeff he'll STYLE you...), visit my buddy Cristophe, the chef at Baracuda(tapas and wine in Hanalei...), hit the Mediterranean in Hanalei right on the shoreline for a cold beer and pupus. Poipu side will be hotter and with less space at their specific beaches. Many guests like that resort side due to the guarantee of hot days and sun galore. The guests i've helped love the rugged north shore with assorted and plentiful beaches that you can just pull off/ up to and wander,snorkel, lay around under a false kamane tree. It depends what resonates with you...specifics like a day up to Kokee and the overlooks of Na Pali are epic and hiking a trail or two around Waimea canyon rim are a must and very approachable. Expansive Polihale beach is wide open yet molten lava hot during the summer but worth an early day trip for a visit..Kilauea lighthouse is a gorgeous view plane, the northern most point in the inhabited hawaiian islands...spend time just flowing and being spontaneous is the key i try to promote with guests; always fun to have an itinerary for specific guest favorite adventures like Princeville ranch zip line or horse back ride due to needing reservations BUT seems many try and cram too much...slow motion savor...get up early and head down to Ke'e and hike to Hanakapi'ai beach and waterfall...i've guided dozens of fun loving folks down that trail showcasing the edible indigenous plants, artesian springs safe to drink from, random rotten guava fights with other hikers, hawaiiana history of the trail...the thrills! I worked the coast for years leading kayak trips, catamaran crewing, outrigger races and intimately know the coast since kid days...depends on the day(wind,if theres an out of season swell, tides,); IF you take capt andys from west side, going into the wind first part of the trip is rough but after snorkeling, you cruise back to port with minimal chop as you are going down wind. Hanalei leaving tours go with the swell and wind all the way down the coast and are way more scenic,lush, spectacular YET coming back as the trades fill in leads to a rough return...might as well go with which ever side you decide to stay on and save the driving.
anyways, long winded...plenty of off the radar and adventures galore... have a good idea of what you think you may like to do/see/savor and allow for spontaneity...always around to kokua
have a wonderful honeymoon! a hui ho! see you soon

Since you have ten days, i think it is reasonable to think about splitting your stay between the north and south. Kaua'i is not that large, but you can't drive all the way around the island. So a trip from Hanalei to Koke'e requires going almost all the way around, then retracing your steps. also there can be congested areas.

If you stay in the area of Anini, Hanalei or Ha'ena ( which is close to the end of the road on the northeast side) then you can soak up the wonderful atmosphere of the North shore. Then move down to the south side, the Po'ipu-Koloa area. You won't lose any more time changing lodgings than you would driving back to the North shore from Koke'e or Waimea area.

There are lots of rentals in all of the areas.

We spent part of our honeymoon in Hanalei, and it was magical.