What services would you recommend to ship belongings to graduate school beforehand?
When I moved coast-to-coast for graduate school, I used


UPS for my personal belongings (e.g., clothes, beddings...etc)
USPS Media Mail for my books and CD's
Southwest Airlines free check-in bags (they allowed up to three pieces back then)


PS: I didn't have any furniture to ship.

If there's stuff you won't need for grad school, send them home until you settle down somewhere with a permanent residence. It'll be cumbersome and costly to haul them around the country.

As for storage, I contacted the department secretary and asked if they could store a few boxes for me (and they kindly agreed). You may also write to fellow grad students in your department and ask if they could receive your boxes for you and store them in their office.

Hope this helps!