I'd say bring along your computer - why bother buying and setting up a new one and leaving the old one unused at home to get older and slower?
You can get connection for your mobilephone after a couple of days as far as I know (Emirates will provide a letter so you can apply even before you have your residence-visa).
Internet - might be faster now too (sorry - no exact info) - used to be once you have your visa. But otherwise there is always the possibility of "Dial n'surf" - a little more expensive (9 Dhs compared to 2 if you are signed up) - but workes pretty well.
Don't bother to bring to much hairshampoo, shavingcream, toothpaste ... as most common brands are available. (don't laugh - I did. Having no clue how Dubai is before I came here...).
Have fun - have a look - and then you'll know!
CU!