The Tourist office (Av. Belgrano and Buenos Aires, Tel.#03414802230, www.rosarioturismo.com) is full of useful information. It is located very near to the Monumento de las Bandera and a trip there is a great way to get started in Rosario. Another branch of the tourism office can be found at the Omnibus station.
Banks are everywhere and easy to find. Up and down Calle Cordoba they can't be missed and on the corner of Cordoba and San Martin there is a particularly dense cluster including a Banco de Santa Fe, a Western Union (where currency can be exchanged) and a Banco Nacional de Argentina. Almost all banks have 24 hour ATMs.
Internet access is everywhere in the form of internet cafés and WiFi is also all over the place. Public telephones can be found on the main drags and in "Locutorios" which are located in every neighborhood.
Every area has their own local laundry services, so it is best to ask staff in your hostel or locals where the closest one is.
The post office, Correo Central (Cordoba 721, Tel.# 03414217226) is located right across from the Catedral on the main drag.
The Hospital Española (Sarmiento 3150, Tel. # 0341 4858700) can fix you up with any medical troubles and reference private physicians if necessary.