You'll find plenty of excellent lunch options near the park, including the Eiteljorg Museum Cafe, inside the Eiteljorg. Another excellent choice is Goose The Market, which is a short drive north of downtown but well worth the trip for the acclaimed sandwiches and exceptional artisan cheese and charcuterie plates (you can also sample and buy wines). You could also drop by the market earlier in the day and pick up a picnic lunch to savor back at the grounds of White River State Park.
After lunch, head into the heart of the city. Indianapolis is the nation's second-largest state capital, and downtown is anchored by the imposing 1888 Indiana State House - on weekends, you'll have to be content admiring the grounds, but free tours of the interior are given on weekdays. For a particularly memorable view of the capitol, head to nearby Monument Circle and its 284-foot-tall Soldiers And Sailors Monument (there's an observation tower at the top, and yes, you can reach this using an elevator).
Downtown - as the hub of politicos and business power-brokers - abounds with superb restaurants, including the wonderfully atmospheric St. Elmo Steak House, which has been grilling hefty chops since 1902 and is justly famous for its eye-wateringly spicy shrimp cocktail. Tastings Wine Bar and Bistro, with more than 100 wines by the glass and excellent food for pairing, and Bluebeard, are a couple of other strong candidates for a fine meal. Or for a more casual and festive experience on the northern edge of downtown, gay-owned English Ivy's is a terrific spot to begin a night of bar-hopping, and Downtown Olly's is a friendly little sports bar and restaurant that's long been a favorite with the LGBTQ community.
Wherever you dine, Saturday night is ideal for hitting the gay clubs on the north side of the city. Hot spots around here include Greg's, one of the city's longest-running gay nightspots and a popular place for watching videos. Neighborhood joint Tini is another great option.