Your residency will be tied to the Emirate where your employer is located, not the emirate you live in. So if you live in Dubai but work in Sharjah (or vice-versa) carefully check the website of their Department of Naturalization and Residency.

Remember you can't make any contract over 30 days until you have residency, so you'll be stuck in a hotel without ability to sign a housing lease. Be sure to get all necessary attested document originals to your dept. ASAP, plus have them schedule a government medical certificate (i.e. no TB, no AIDS).

Getting your Residency visa is your employer's responsibility: see Residency link on Mobilization for details. Getting a spouse's visa is your responsibility as their sponsor, so do check out DNRD and SNRD websites.