@ quentinlittle, if you're got a few days or a week in Dubai on a backpackers budget, then I guess the best thing is to stay at the hostel in Al Nahda or one of the cheap hotels in Deira. In terms of things to do that won't cost you much, you could easily pass a day or two exploring both sides of Dubai creek. this is the original hub of Dubai so there are quite a few souks (Gold souk, Spice Souk, Textile souk, Fish market etc) and lots of dhows loading and unloading.
There is also the Bastakiya area which is traditional, where there are a few good cafes and some art galleries which I think are free to enter. Then there's Dubai museum nearby Bastakiya and Sheikh Saeed's house etc. There are some interesting little lanes off Al Fahidi Street which have a bit of an atmosphere.
Crossing the creek is cheap and easy on the abra (small boat - cost is about AED1 I think).
You could also spend a day just messing around on Dubai metro - gives you a good feel for the city. A one day pass is only AED16 and I think that includes the buses too (not sure).
Have a look at the Burj Al Arab, go to the public beach in Jumeirah (free) or the Jumeirah Beach Park (AED5 entry). There's Palm Jumeirah but it's little more than a very expensive housing estate.
Go and look at the Burj Khalifa (world's tallest building), they don't charge you for looking at it. You can also watch the dancing fountains at the Burj Khalifa at night which sounds rubbish but it's actually fairly good. Or am I just getting old? There's ice skating in Dubai Mall and also an aquarium - the latter is free.
Have a schwarma and a pomegranate juice at one of the many street cafes in Deira or Bur Dubai. Try Cheese manakeesh from the Lebanese Bakery in Karama. Go wandering around the shops in Karama selling fake watches, clothers and handbags.
Outside of Dubai, consider a trip to Sharjah for souks, the waterfront, a few museums and more of a traditional Islamic atmosphere. If you're staying in Al Nahda, you are anyway on the Sharjah border.
There are the mountains and wadis in Ras Al Khaimah but I don't think it will be easy for you to access them without a car unless you take a tour. There's also the desert but again it requires a car or going on a tour. On the east coast there'a a bit of snorkelling and diving around the Dibba area, plus a few old forts and mosques. You can probably get a bus over there but budget accommodation is generally quite difficult to find - it's mainly international hotels.
My suggestion is just spend 2-3 days here if you're backpacking, there's not enough to warrant staying longer unless you just want to sit on the beach. One book worth reading is 'Hello Dubai' by Joe Bennett - he covers all that good bits and the bad bits of Dubai.