Avoid the chain eateries at all costs!!
Birmingham has so many independent shops and restaurants.
Ok best coffee - 6/8 Kafe round the back of house of Fraser. No baby changing there but it's a very laid back place (small and gets a but busy, maybe a bit of a squish with a pram but they have an outdoor area). HoF has amazing bf and baby changing facilities.
Urban coffee (Church st) and Yorks Bakery cafe are nearby, have amazing coffee and vibe; baby friendly (lots of mums and prams at Yorks) and you are near enough HoF to use their changing facilities. More pram space than 6/8
Short walk away is the new library, gets a but busy on weekends but is a must see. Great gardens, terraces and children's section (where you have excellent changing and bf facilities). You can also use the library or the symphony hall (nearby) to change/bf if you do a canal tour (you can just amble around, or take a boat)
On the canals, worthy of note are the Canalside cafe (cafe/pub cross), JuJu's (lovely bistro, open late afternoons but not hugely baby friendly though they have a terrace). Tap and spile pub or the Prince of Wales pub for your DH. 
By far the most family and baby friendly restaurant is Cafe Soya. Yes you can go to Jamie's and the like in the Bullring, but a two min walk from there, you will find Cafe Soya. They make amazing, fresh and cheap Vietnamese, Thai and Chinese food, lovely staff, all round great place with loooooads of space for prams and a baby changing area. I love this place!
Other Brummie favourites in the city centre are Asha's and the Bodega.
In the jewellery quarter, be sure to check out the Church Inn pub, or The Lord Cliffden or the Red Lion (another one next to the JQ clock - the Rose Tavern I think). All have excellent good and are a bit of a hipster/yuppie hangouts, very relaxed, but not sure what child policies they have!
Urban coffee also has an outpost in the JQ
The Ikon gallery (just off the canals) is worth a visit, lovely cafe too. Sometimes they have children's events.
The Mac arts centre, just a short taxi ride out of the city centre, has loads of family and children's events on. Nearby Canon Hill Park is good for a walk.
Ah, I'm so glad my comprehensive knowledge of Birmingham came in useful at last!