The Hyatt is downtown - they have a little shopping galleria where the Hyatt is and close to it. And you're in downtown - there are a few shops there that can be kind of fun. Some homeless as well, unfortunately. Find Central Avenue and you'll find lots of little shops - it's only a few blocks (if that) from your hotel - really close. Some neat restaurants, bars, a movie theater.....
There used to be a "trolley" that went to old town and back, so you might ask concierge about that - if you are a good walker - it's about 15 city blocks. Don't know how much of a trek you want to make - to old town. Botanical gardens is further than that. They do have taxis, and the Hyatt is a nice hotel, so you shouldn't have any trouble finding one. They are pricey, as I recall.