When you arrive it is very simple to purchase a local SIM card with a Qatari phone number. The international access code for Qatar is +974 and by adding a + to your home country code you can call home quite affordably from your mobile.
The country is well covered by the GSM 900/1800 so your tri-band phone will have no problem finding a signal to keep in touch wether roaming on your home plan or using one of the countries two carriers that offer 3G service. There are also options for USB internet modems as Qatar boasts a complete HSPA+ that is one of the fastest in the Middle East.
For those traveling without a laptop (really?) there are numerous Internet cafes around Doha.
And last but not least, Q-Post handles the country's incoming and outgoing mail services and is relatively inexpensive an reliable for sending things to Europe and North America. But should you prefer FedEx, UPS or DHL they are all in town as well.