I had the same problem with Tiger, and was advised by the paediatrician to first cut out completely as it's a surer way of telling whether or not you've hit on the right problem, then to slowly reintroduce after a few weeks (to make sure it really is out of yours and baby's system first) and see if he can handle you having small amounts.
Obviously if cutting out dairy doesn't solve the problem you'll have to dig deeper, but it's completely resolved it for Tiger, and I find I can have a cup of tea with milk in, and a few bits of chocolate, or a slice of pizza, or something equally "milky" a day before noticing any issue in him.
Re. introducing it into baby's own diet - Paed advised keeping him off dairy completely until about 18 months (I know advice on this age does vary from country to country) and then start slowly, in the order ratoncito said: milk in cooking or baking, cheese, yoghurt, and then "drinking milk". If it's a no-go, try again in 6 months and then every 6 months thereafter until no symptoms appear. I had the same problem as a baby, and could tolerate dairy at about 4, though I still can't overindulge...
Mum to two gorgeous boys, Bookworm (May 2009) and Little Tiger (November 2011)
Musings about our homeschooling journey at www.smallstepstoabiggod.org