We go to St. Augustine every year & consider it our 2nd home! Some day, we hope to retire to the area.
For one night, I'd suggest maybe one of the many bed & breakfasts in town. There's also a few hotels downtown - some facing the bay. However, if you wanted to stay on the beach, the Hampton Inn or Fiesta Inn on the beach are nice - they are both on the beach side of the road.
If you're only planning one day, I think the Fort is a must-do.
The Pirate Museum is also really well-done. As someone else mentioned, it's designed by former Disney Imagineers. And it's right across the street from the Fort.
You could tour the Fort in the morning before it got too hot, cross the street & tour the Pirate Museum, eat at Spanish Bakery for lunch, shop on St. George Street for a while, eat at The Columbia for dinner, & then get an ice cream cone & walk along the bay in the evening.
In addition to drinking from the The Fountain of Youth, you can also learn a lot of the early history of the area. Plus, peacocks roam the grounds. Lightner Museum & Ripley's are also good. (And if it's sweltering outside, the Lightner Museum & Ripley's are inside.) The Old Jail is fun, but not as macabre as it used to be - they've toned it down quite a bit.
The Original Ghostly Walking Tour is also wonderful. There are TONS of different ghost tours, & some are really cheese-y. But the Original Walking Ghostly Walking Tour combines actual historical events w/ ghostly legends of the city - so it's a neat history lesson too.
St. George Street has lots of interesting little shops & restaurants, & there are side streets & alley ways off St. George Street w/ more of the same. Some of the shops & restaurants do tend to close early, but, during the tourist season, more & more are staying open later. (St. George Street is pedestrian only traffic.)
Parking is not great. You can park in the garage, but it's pricey. Sometimes, you can also find parking at the Fort. If you're staying at one of the bed & breakfasts or hotels in town, you won't need to worry about parking.
The Bayfront is a lovely area - often times, you can see dolphins in the bay. There are several different boat tours. This past June, we did a kayak tour in the bay.
For lunch, Spanish Bakery on St. George Street is delicious!
There are several different little shops that offer sangria tastings, but you can't beat the sangria at The Columbia.
For dinner, we like The Columbia, Harry's, and Barnacle Bill's. Aunt Kate's & Cap's on the Water near Vilano Beach are good too. On the beach, Sunset Grille, Oasis, & Saltwater Cowboy's are good. My parents also love the little Wildflower Café.
Our kids enjoy the Alligator Farm, & the Lighthouse is great too - you don't have to climb to the top (the view is great though!) & can still tour the lightkeeper's cottage & the grounds.