You have your choice of Jacksonville Beach, Neptune Beach, or Atlantic Beach.
One dog-friendly hotel at Jax Beach is Quality Suites.
Welcome to Quality Suites Oceanfront, a Jacksonville Beach, FL hotel.
IMHO Jacksonville is very dog friendly (not the most dog friendly, but definitely better than some cities), and you'll find there are many hotels that welcome dogs. Some restaurants with outdoor seating allow dogs there, too. Vito's at The Landing (downtown) has in the past. Starbucks in San Marco does.
Dogs are allowed on Neptune and Jacksonville Beaches before 9am and after 5 pm, and any time on Atlantic Beach. Be sure to have documentation about licensing and, more importantly, an up-to-date rabies tag and certification.
There are also a few places in Jax Beach to rent bikes. Here's just one...
Rent Beach Stuff Jacksonville Florida : Bicycles and Accessories
For artsy places, go to San Marco and Five Points. Riverside and Avondale also have some interesting places. Additionally, there are some galleries downtown including the MOCA (Museum of Contemporary Art).
A nice day trip would be a ride up A1A, taking the Mayport Ferry and continuing northward to Amelia Island and then Fernandina Beach. There are a number of state parks along the way. Dogs are welcome in most of the parks, but not on any of the state park beaches.