G20 starts then after driving, go into the store, no start, replaced battery

Numerous things have to work in order for car to start. Starter relay, key switch,nuetral safety switch in shifter linkage, brake switch, had problems with 2001, exactly like you. Take it to someone you can trust or a dealer, in ft myers fl the cost was low, had them all replaced. Mine turned out to be key switch after replacement had locksmith rekey cylinder to original key.