Might be worth hopping on a bus and going to the taste of Birmingham festival at Cannon Hill park - a child friendly event too.
Lunchwise, Martineau Place (opposite Rackhams) has the usual lunchtime places to eat e.g. Eat, PIzza Hut etc. plus it's much less busy than the places near the Bullring. If I were you, if it's nice weather I'd grab a sandwich and go up to St Phillips Square (other side of Rackhams/House of Fraser) and eat outside by the church.
Shopping wise I'd definitely recommend some of the smaller independent shops in the Great Western Arcade and the Picadilly Arcade. If you like silver jewellery, Smithsonia in the Picadilly Arcade (just off New St towards the top end) has saved me from many a birthday dilemma.