I shall be visiting and staying with some friends in Dubai, and want to take a day trip out to visit another group of friends in Abu Dhabi. Without hiring/driving a car, what's the easiest way to get from one to the other and back for under 100 USD? According to Wikitravel, a taxi from Dubai seems to only be about 300AED (~80 USD) - are taxis really that cheap in the UAE?
The easiest way is indeed a taxi, and yes, that price sounds about right, although it's a regular metered fare so the exact cost will vary. Not every taxi will be willing to up and off to Abu Dhabi on a lark though, but the airport taxi stand can sort one out if you ask, or you can call and book a cab. As for safe, well... most UAE cabbies drive fairly sanely by the standards of the region, but the Dubai-Abu Dhabi highway always has the occasional blinged-up rich kid doing 300km/h in the fast lane in Daddy's Lamborghini.
The cheaper and safer, but slower and less convenient, option is the Emirates Express bus, Dhs 20 one-way. Buses leave from Ghubaiba station, on the Dubai Metro Green Line, every 30 min or so.
Taxis are that cheap in Dubai, and in general the public transport system is one of the best in the middle east.
Your options for Dubai - Abu Dhabi:
Public Transport (routes, timetables, etc. available at this website). The first bus to Abu Dhabi leaves at 5 AM and the journey takes about 90 minutes given the traffic conditions.
Careem - this comes to $110 but it is the most comfortable and safe way to go. I would recommend this over all other options.
Flydubai has a bus service that you can book as well. Its 40 AED one way; from the airport to the airport.
Just to add up, a bus from Dubai to Abu Dhabi would cost much less. There are two buses E100 and E101 that take you from Al Ghubaiba Bus Station (Dubai), or Ibn Battuta Metro Station (Dubai), right to Abu Dhabi Bus Station.
If you're settling in Dubai check DubaiHow, they actually have the routes of each bus in Dubai in details without the hassle of maps and some out-dated Wikiroutes: https://dubaihow.com/dubai-bus-routes/