I am in St. Augustine about 6 times a year for work and enjoy the area.
Castillo de San Marcos, the old Spanish fort, is a national park and a must see in Old Town. Extremely cool when you take the time to do the tour and learn about the fort and the area.
There are other tourist-type things there- Fountain Of Youth, Oldest Schoolhouse in the USA, etc. but I’ve never done those. Most people mill around the Old Town area- St. George Street and the like. Plenty of shops and restaurants. There are some good spots there to enjoy adult beverages and some good food. Meehan’s is a good Irish spot, Prohibition Kitchen is good for a nice lunch and a drink. I keep hearing about a bar/restaurant called the Ice Plant Bar that is supposed to have great craft cocktails and a good menu but I haven’t tried it out yet.
As for lodging, I typically stay at the DoubleTree just north of Old Town and the rate is usually pretty decent at around $150 this time of year. Stays there include a decent breakfast in the morning which could help out on expenses. They have a free shuttle bus that will run you the half-mile to a mile to Old Town and the fort.
Hope some of this is helpful.