Depending on the weight and size of a letter or parcel the costs are different. In Germany we call this "Porto" (postage). You can buy stamps in any post office (Deutsche Post) branch or at kiosks. If the words "Porto zahlt Empfänger" are written on the envelope, no stamp is needed.
If you have a very important letter, you can send it as registered mail (Ger. Einschreiben). The recipient must confirm the receipt with their signature. While registered post must be handed in at the post office branch, other letters may be put in the yellow letter boxes found in every neighbourhood.
The company responsible for sending parcels through Deutsche Post is DHL. There are other courier companies available, such as UPS and Hermes.
Most flats already have a telephone / TV / Internet connection. To use it, ask the landlord which provider is most suitable or research the best package according to your use. Most packages offer flat rates for the landline (Ger. "Telefon Festnetz"), TV (Ger. "Fernsehen") and internet connection. Mobile phone (Ger. "Handy") packages are also available. These can be ordered in a branch of the provider or online, and contracts are mostly for a fixed period of time. To compare prices, search for "Preisvergleich" and the type of package you are looking for.
International calls are usually not covered by a telephone flat rate. It is cheaper to make your calls over the Internet (e.g. on Skype or Whatsapp).