Sudbury has a good selection of traditional Suffolk pubs, several of which are genuine locals rather than tourist operations. The town sits in the heartland of Greene King territory, and you will find their ales — and those of smaller local brewers from around the county — on handpump throughout. Market Hill and the streets immediately surrounding it are the best hunting ground.
For coffee, there are a handful of cafes in the town centre, mostly independent. The quality is decent rather than outstanding — this is not a destination for specialty coffee, but you will not struggle to find somewhere to sit with a cup and watch the town go about its business.