Getting around the United Arab Emirates
I will mainly elaborate on Dubai since I have not set foot outside. I would just say that to move from one city to another, apart from taxis, you have public buses which are rather cheap by the way.
Dubai – Abu Dhabi
A trip from Dubai to Abu Dhabi is 25 AED (approximately 6 €). There are 2 lines, the E100 et E101 the first of which ends in the south of Dubai while the second goes up to the north of the city.
Dubai – Ras Al Khaimah
If you want to go to the northern mountainous area, in the emirate of Ras Al Khaimah, there is a shuttle bus from Dubai airport (terminals 1 and 3) to promote the area. There are 6 buses per day between 6am and 1am (so not many people). It costs 20 AED one way and normally takes 45 minutes.
Dubai – Fujairah
The other mountainous area, located to the east of the emirates, with enclaves in the heart of Oman, its neighboring country. You have to take theE700 which departs every hour from Dubai's Union Square Metro Station and stops at the Dubai bus terminal Fujairah 2h later. Price = 25 AED.
Dubai – Ajman
The smallest of the emirates, Ajman, is located north of Dubai and should set you back AED 12 for a 40-minute journey. It operates daily between 4:30am and 23:30pm (weekday hours).
Getting around within Dubai
Since 2009, you can use the map NOL (Christmas Card), a rechargeable card that allows you to easily use all public transport in the city of Dubai. This also applies if you take the intercity public buses that I just mentioned.
It is practical to avoid having to take out cash every time you go somewhere and it is also possible to associate this card with your smartphone to pay directly with your mobile. (subject to compatibility). Afterwards, I suspect that this solution is especially advantageous for people staying longer on site.

Dubai Metro Map.
The metro in Dubai
There are currently 2 metro lines in Dubai, partly above ground and partly underground. The main one, the red line, is the one that starts from the international airport and runs through the city from north to south. The other, the green line, makes a sort of serpentine in the "historical" part, to the north of the city.

An underground metro entrance.
It is very convenient and since the city is very long, it allows you to access different areas easily. Plus, it's really not expensive since it costs less than 5 AED the journey.
However, the city being a certain width, the metro does not allow you to reach all the major tourist sites, I am thinking especially of Burj Al Arab. It's a bit of a walk, but as it can be very hot, it's not necessarily the best option. Then there is the taxi.
Taxi in Dubai
Again, I was surprised at how affordable these were. Expecting everything to be expensive, the The support rate starts at just AED 10To give you an idea, a trip from the international airport to the Marina should cost around 80 AED, or just 20 € for 35 km.

A view of Dubai from the taxi.
Buses in Dubai
There is a public bus system. The special feature is that, given the summer heat, there are air-conditioned stops. It's not expensive either, from 1 to 4 AED the journey. But since I would have been too dependent on schedules, this was not a viable option for my express visit.
Note that there is a bus open-tour, a typical tourist bus, I will take you around the city with the main sites (hop on hop off). You take a daily package (220 AED anyway) and go up or down wherever you want. The bus route navigates between the city's important spots with a choice of 2 possible circuits. There is also a bus that runs at night to admire the lightings from Dubai.

Air-conditioned bus stop, only in Dubai!