I don't get out much so I can't comment on too many things to do.
We have Lake Shawnee on the East side of town, not too far off of HWY 70. Aside from that and local historical landmarks, Capitol, Brown vs Board, etc..., I can't recommend any hot spots. I just don't know of any. Lawrence, Kansas is just 20 minutes East and is full of great food and drink.
Here is what I recommend for food in Topeka.
Blind Tiger Brewery and Restaurant (I do love, love, love many of their beers but their food is consistently missing the mark, IMHO. But the beer... Man, go here.)
RowHouse Restaurant (Pricey, great food, only open for dinner Wednesday through Saturday.)
A.J.'s Pizzeria (You guessed it, pizza. This place has never left me disappointed. Tallgrass Brewery cans available, can't remember any other breweries served.)
The Burger Stand (A great gourmet hamburger joint with a full bar. Also has Blind Tiger and Free State Brewery on draft. My favorite place in town.)