One of the main things to do in Baton Rouge is to eat delicious food.
Enjoy exploring Louisiana Creole cuisine (surprisingly great Wikipedia article!) and Cajun cuisine. Two favorites: blackened redfish, and bread pudding.
Eat the incredible seafood poboy (get it with sprouts, and hashbrowns on the side; apply tabasco liberally) at Louie's by LSU (open 24hrs, usually--closed sunday nights?).
Be awed by the epic summertime thunderstorms that roll through almost every day around 2pm.
Visit the observation deck at the top of the state capitol. It's open till 4pm. Prepare for your visit by reading All The King's Men by Robert Penn Warren (or watch either of the films--the 1949 film won best picture, and the 2006 re-make was filmed locally), a fictionalization of the rise and fall of Huey Long. Pick up a copy of the book at Cottonwood Books.
Two local obsessions: Raising Canes chicken fingers and LSU Football.
Eat pizza at Capitol Grocery in Spanish Town, at 5pm (except Sunday). Sit outside and listen to some locals telling stories. Wander around Spanish Town and Arsenal Park.
Run/bike/drive around the LSU lakes. Gawk at the amazing houses.
Eat delicious food at George's restaurant, an incredible dive bar under I-10. Favorites are the burgers (the 'heavy hitter' with avocado), the pastrami and swiss on rye, the ribeye sandwich. Legendary for their shrimp poboys, though I have never had one. Leave a dollar on the tar-encrusted ceiling.
Play tennis or golf at City Park or visit the dog park
Drink beers, eat red beans and rice, boudin balls, and hushpuppies at the Chimes by LSU. Tin roof amber is a great local beer (it's not on the menu, but they have it!). If it's your first time, start out with an Abita Amber and a fried alligator appetizer.
Admittedly it isn't Cafe du Monde, and, after being razed by Walmart, the neighborhood ain't what it used to be, but you can still get your beignet fix at Coffee Call.
Visit the new Tin Roof brewery (friday afternoons only) and enjoy free samples.
See a show and get dinner at Chelsea's, also in the I-10 overpass area. One favorite is the grilled cheese on foccacia; goes well with a blue moon.
Drink coffee at PerksGarden District Coffee (on Perkins Rd) or Highland Coffee (by LSU; always full of lots of studying students).
Get a plate lunch at Zeeland Street Market (by Perks). Get the lunch special. On Wednesdays they have the best fried chicken in town. On Fridays get the fried catfish with mac and cheese on the side. Best time to arrive is just before the 12:00 noon crowds. Closed Sunday.
Take a date to lunch at Yvette Marie's, a cute low-key restaurant in an antique store. I like the jalapeno chicken sandwich. If you're looking for something more traditional, you can't go wrong with their muffuletta sandwich.
Ride in the monthly Critical Mass bike ride with approximately 200 other cyclists through the streets of the city. Last friday of every month, 6:30pm, LSU parade ground/clocktower. See also the bicycle events calendar.
Some other restaurants to look up: Parrain's Seafood; Juban's; Roberto's River Road Restaurant (Sunshine, LA)
The Old State Capitol is beautiful, historic, and free to visit. On the river at North Blvd (by the Shaw Center).
Stroll on the levee and watch the ships (barges) go by.
If you are a civil engineering / geology nerd, you will enjoy reading John McPhee's book The Control of Nature (or read it online) which details the century-long but almost-certainly-doomed effort to control the Mississippi river. If this stuff interests you, drive up and visit the Morganza Spillway and Old River Control, about 1 hour drive north from Baton Rouge (maybe a bit shorter now due to the new Audubon Bridge). There is also the Bonnet Carré Spillway on the way to New Orleans. (Morganza is also the location of the "cafe scene" from Easy Rider; visit The Bear (bar) for some memorabilia.) Check out this beautiful overlay of some old geological maps showing the past courses of the Mississippi river onto Google Maps. Roadside Geology of Louisiana is good too.
The Laser Interferometer Gravitational-Wave Observatory is about 30 miles east and offers public tours on some fridays and saturdays. Contact them in advance. CAMD operates a synchrotron light source in town (across from Whole Foods); you might be able to get a tour there too.
Get a group of friends together, bring a cooler full of beer, and go Tiki Tubing down the Amite River. If Tiki Tubing isn't quite your style, rent a kayak at the Backpacker and take it out on some local river or bayou. They have equipment that will let you carry a kayak on just about any vehicle.
Head out to Zydeco Breakfast at Cafe des Amis in Breaux Bridge (1 hr drive west) early Saturday morning (8am). Or the cajun/zydeco dance at Whiskey River Landing Sundays at 4pm, or their neighbor McGee's Landing Sundays at noon (also: airboat rides). Listen to KRVS 88.7 FM on the way over.
Tour Laura Plantation and stroll the grounds of Oak Alley Plantation. I've heard Laura Plantation has a much better, more historically-informed tour; skip the tour at Oak Alley and go directly for the mint juleps.
Get an airplane flying lesson at Fly By Knight (Hammond)
Go to Tsunami on the roof of the Shaw Center (art museum) for the best view of the river (thanks BiscuitCrisps). Great place for a drink! Also, check whether any events are going on at the Shaw Center or the co-located Manship Theatre. They often have interesting shows and films.
The Cove has this city's best selection of whiskey (thanks malakhgabriel).
One of the main things to do in Baton Rouge is to eat delicious food.
PerksGarden District Coffee (on Perkins Rd) or Highland Coffee (by LSU; always full of lots of studying students).