Downtown Gainesville has a beautiful (really beautiful) public park at the Thomas Center. Originally a home, then a hotel, then a Santa Fe College campus, and finally a museum and City offices, this building is beautifully maintained inside and outside.
I love to walk under the beautiful, old live oaks. Try to find an event as well- NIA classes are offered regularly, as well as yoga.
These 2 historical rental homes are just 3 blocks from The Thomas Center: https://www.gainesvillecorporatehousing.com/houses-1-2-3-and-5-bedrooms
For NIA classes, please visit: https://www.facebook.com/NiaGainesville/
For yoga classes, please visit: https://stayhappening.com/e/outdoor-donation-yoga-tues-andamp-thur-E2ISTGSK1UO
For more information about The Thomas Center, please visit: http://www.historicthomascenter.org/