The United Arab Emirates, also known as the UAE consists of seven smaller 'emirates' which can be likened to 'states'. Dubai and Abu Dhabi make up two of those seven emirates and are the largest cities within the country. Therefore, it's no surprise that these two cities are also the most popular cities for visitors - hence visitors tend to take advantage of that fact and travel between the two cities!
The Distance Between Dubai & Abu Dhabi
According to Google Maps, the distance between the two cities is roughly 140 km with an approximate travel time of an hour - but that, of course, depends on where you're coming from and where you'd like to go as it can even take you up to 3 hours to travel from city to city.
Dubai to/from Abu Dhabi Via Car
One of the best and most flexible ways to travel from city to city is via car. Rentals cars are reasonably priced and not to mention petrol is inexpensive - one of the benefits of living in an oil-rich country. If you'd prefer to go to "the beat of your own drum" rentals are certainly the way to go as you'd certainly have greater flexibility and free range of movement.
Be sure to travel outside of rush hour as the traffic can be pretty intense.
Dubai to/from Abu Dhabi Via Bus
The cost of a one-way ticket between Dubai and Abu Dhabi is 25 AED. making this means of transportation the most affordable way to travel from city to city.
If you're travelling to Abu Dhabi, you'd have two options as the bus service serves two different parts of the city - buses will arrive and depart from Ibn Battuta and Al Ghubaiba Bus Station.
If you're travelling to Dubai, the bus departs from the Abu Dhabi Central Bus Station.
Dubai to/from Abu Dhabi Via Taxi
Probably one of the most expensive means of transportation - this isn't an option that I'd recommend due to the distance and time it would take. The cost from city to city ranges from 200 AED to 300 AED and the rates of Uber tend to fluctuate. I'd only recommend taxis/Ubers for short distances.
Dubai to/from Abu Dhabi via Airline Coach Service
Both Etihad and Emirates Airlines offer a free coach service for economy customers traveling to/from Abu Dhabi International Airport from/to Dubai.
Be sure to book your spot on these coaches at least 24 hours in advance.
Truthfully, if you prefer not to drive, the best option would be to book a tour that has designated pick up and drop of points. We traveled from Dubai to Abu Dhabi twice - one with a friend who lived in Dubai and secondly with a tour company. With the tour company, our pick up and drop off location were within walking distance to our hotel and at least we were guaranteed to see the major points of attractions within the city of Abu Dhabi. This seemed to be the easiest option as we traveled with a large group of persons. On the other hand, when we traveled with a friend it was more intimate as they weren't as many persons as we were used to traveling with which gave us a lot more freedom and we had to opportunity to explore Abu Dhabi from the eyes of a "local".
In the end, there are a lot of factors which would affect the mode of transportation that you require but it's not hard to travel between the two cities and you definitely should!