For a traditional experience check out Gem's, for newer and more fun flavors try The Bagel Nook
Patricia's in Holmdel is great, but for a more authentic and fancy experience try Assaggini Di Roma in Hazlet (both are BYOB, literally go to the liquor store before hand and bring some wine or beer with you, I promise its normal here)
If you've heard of Dave Portnoy he's reviewed a lot of local places - feel free to take his recommendation. If not, we recommend Naples Pizzeria in Belford (good for take out or sit in, also BYOB)
For a more casual experience try On the Deck in Atlantic Highlands (it has a GREAT view of NYC, and is the first place Amanda and her dad tried when they came out to visit before she moved here), for a fancier dinner try Bahr's Landing in the Highlands
Downtown Red Bank has lots of restaurants and shops, plus the BEST arcade ever filled with retro games that everyone will love (check out Yestercades!)
Even though it will be cold, Sandy Hook has some beautiful beaches and a light house and museum to explore
On a clear day, Mount Mitchill has a beautiful view of the bay and the city
One of the coolest buildings and community spaces you will ever see, plenty of restaurants and shops, but at the very least worth a look around (the show Severance was filmed here, and it has a very cool history with Bell Labs - ask Amanda about it!)
The cute shore town where Amanda used to live and where we got engaged at sunrise on the beach (drive down HWY 36 and look at all the massive beach houses on your way). Lots of restaurants and some shops right along the beach
Lots of restaurants, Steve's favorite ice cream parlor, and the marina