Since you posted in the Newport forum, I'll assume that's where you will be visiting.
There are several fresh fish markets: OceanBleu, Fish Peddlers, Local Ocean, and South Beach Fish Market. The last one is in South Beach south of the bridge; the others are on the bayfront.
Purchasing fish directly off the boats is possible. The F/V Chelsea Rose (also The Barge) sells as do some others. Usually you will just see small signs around Port Docks 3-7, at the bayfront.
When you say "bring it back home with me" do you mean to wherever along the coast you'll be staying...or Arizona? If the former, no problem. If the latter, you should talk to the larger seafood operations along the bayfront to see if they'll flash freeze it and ship it for you or overnight express it on a future date. If you think you'll just keep it in a cooler and drive it home or check it as baggage when you get to an airport...well, it won't be fresh by the time you'll be ready to eat it.
Some of the charter fishing ops do that for their anglers. Maybe you can work something out with them.
What is the best time of the day to be at the docks to buy fresh catch of the day? we'll be there in 2 weeks - getting more + more excited about the fresh catch seafood! Thanks!
No set time because there are so many variables. Look for a boat that has a banner proclaiming "Fresh Fish" or similar or sign on the dock directing you to a boat.
If you want a real local treat get a place that deals with local catch, they're rare, and keep an eye out for Ling Cod. The most notable local catch is Dungeness Crab and there is a lot of locally farmed oysters. Dishes made with smoked salmon is a Northwest tradition although the salmon is mostly Alaskan.