For getting registered in the local library, normally photo ID and proof of residence is required. You can show your passport and copy of lease to become a member. Check with the library if they require anything else.
One reader has asked "Should we search for public schools in possible location we would be moving or should we first look for a good school-district and then accordingly select a place / city of residence?"
I think you should first search for public schools in the area where your office is located. Then try nearby areas if the rating is not good and accordingly look for place of residence. Office proximity is good if you have to rush to the school for some emergency or to attend parent-teacher meetings.
After you have shortlisted the school, you can call/e-mail the school/district and ask them the procedure of admission. Normally they require proof of birth, immunization records signed by local pediatrician and proof of residency (you can get admission only in the school allocated to your current residence). My son had joined school in 1st grade but the school did not ask for any previous school records. You have to check with the school.
In my next blog, I would give information on how to look for apartments in USA and what are the things you have to consider.