While there really is not much to do in Beech Grove, Indiana, there are some great surrounding areas including downtown Indianapolis and Fountain Square. The crime and poverty level is very low. On top of that, Beech Grove is considered to be a part of the lower income class.
Some of the best areas around Beech Grove include Fountain Square and Indianapolis. Fountain Square is a very hipster area connected to downtown Indianapolis that includes some of the best, unique restaurants like Milk Tooth. Milk Tooth is a breakfast scene open only some days throughout the week until about two. (I would recommend checking the website to see what days they will be open the week you go.)
Tappers is also a place in Fountain Square that is popular for its arcades and beer. Including arcade games of Frogger, Mortal Combat, Mario, Gauntlet, and more. It is usually really busy on weekends and you must be 21 and over to enter after ten.