Read and Rhyme is just stories with a bit of music and singing. At the one in Foggy Furze they have little instruments. The central library will be the closest to you. It's just outside the shopping center.
There is a health shop in the shopping center, but I'm not sure what bio foods are? If you mean organic, yes the health food shop stocks certain things and all the supermarkets have organic.
There is a farmers markets at the Marina every 3rd sunday of the month, some of that is organic.
Baby equipment - there are a few stalls in the market and Argos etc. But you'd probably be best driving to Teesside Park and going to the big MotherCare.
Hartlepool is not a city. It is a town and I wouldn't get too over excited about it either. While it can be lovely at times, it's hardly metropolitan and imo has virtually nothing to do and no decent shops. But I suppose how you feel will depend upon what you are used to.
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