Nearly all hotels now offer Internet access at relatively fast speeds and many offer wireless access. There are many restaurants and coffee shops, for example Starbucks, across the UAE that offer wireless access, and sometimes it’s free.
As far as Internet access goes, bear in mind that some websites are banned from the UAE, such as anti-UAE sites, VOIP sites, such as Skype, pornography etc. Flickr is the most notable ban. Some notable bloggers have been banned in the past including the notorious secretdubai blog. As with all technology, there are ways around bans, but that could get you in even more trouble!