As a new expat, you can't even open a bank account until you have a UAE mobile phone number. Etisalat has the majority share of the mobile phone market - the other company is Du

Initially, it has to be a pre-paid phone, as legally you can't enter into any contract longer than your arrival visa validity- typically 30 days. You also need to provide a copy of your passport pages showing your identity and visa.

You can buy any handset at any store, plus SIM card pre-loaded with e.g. AED100  worth of minutes. "Wasel" minutes can be topped up as needed at gas stations, many stores. You'll get a receipt with a long string of numbers - on your phone type  in "120" then string of numbers. To verify the update, type "*121#" and you get back a text message with new balance on the SIM card.

Once you have a Residence Visa, switch to post-paid GSM, which is a cheaper rate. If you travel outside UAE, they ask for a significant deposit to get roaming access, plus it costs several dirham per minute. Better to get a pay-as-you-go SIM card with a local number if you will be outside UAE for a while.