Taxis are more convenient. You hop into one at the curb at the north end of the baggage claim area and tell the driver where you're going, then he takes you there (on a direct route, if he's honest and/or you make it clear that you know which way he should go, you'll only pay how much you know it should cost whichever way you go, and you'll call his dispatcher and the Taxicab Authority if he doesn't) forthwith.
Of course, for the convenience, cabs are more expensive. Just to get into the cab is $3.30, plus $1.80 for the airport pick-up fee. Then it's 20 cents for each additional 1/12th of a mile ($2.40 per mile) and $30 an hour for waiting time.
According to an extensive chart at http://www.taxi.state.nv.us/FaresFees.htm, the cost of a typical ride from the airport to the MGM Grand is $11.58 (based on distance) to $13.58 (including four minutes wait time at traffic lights). To Luxor it's $12.78/$14.78. To Caesars, it's $16.14/$18.14. To the Wynn, $15.30/$17.30. To the Riviera, $18.30-$20.30. To Circus Circus, $21.18-$23.18. And to downtown hotels, $21.66/$24.48.
Airport shuttles are a little less convenient, depending on how crowded they are, but they're also noticeably less expensive. The way they work is you buy your ticket to the Strip or downtown (at kiosk windows outside the baggage claim area), hop on the first passenger van in line, and leave when it's reasonably full; that could take 15 minutes or so. Then the shuttle makes a circuit, generally among the south Strip hotels, the center Strip hotels, the north Strip hotels, the downtown hotels, and the off-Strip hotels. So you might have to make a few stops and let off a few passengers if your hotel is toward the end of the line.
Sometimes shuttles combine Strip and off-Strip passengers. This writer once sat on a shuttle that stopped at Bally's, Imperial Palace, Harrah's, and Caesars before dropping him off, last, at the Rio. The whole trip, from stepping on to stepping off the shuttle, took an hour. But he didn't mind; he got a little tour of the back ways in and out of the joints along the way.
Most of the shuttles (Ambassador/Ritz, Bell Trans, Gray Line/Coach USA, C.L.S., Executive, and Showtime) charge $6-$6.50 per person to the Strip, $8-$8.50 per person to downtown, and $7-$7.50 to off-Strip casinos.
Round-trip fares are slightly less expensive; be sure to call at least two hours before you want arrive at the airport, or better yet, make a reservation 24 hours in advance. Also ask your driver for his suggestions about timing.
Including the higher tip for a cabbie than a shuttle driver, if there are two passengers in your party, the combined shuttle fare is generally competitive with the cab fare, though perhaps a little lower in most cases. If there are three or more passengers, you'll generally save money and definitely save time in a taxi.
