Summarize:

A mandolin is traditionally tuned G-D-A-E, from low to high, with each pair of strings tuned to the same tone. In other words, the instrument is tuned G-G-D-D-A-A-E-E, taking into consideration each individual string. When you're holding the mandolin correctly, the highest string pair (E) should be the closest to the floor. If you play guitar, it can also be helpful to think of it as the lowest four guitar strings (E-A-D-G), but in reverse. That can also help you get familiar with the fingerings when you're first getting started on the instrument. On most mandolins, the tuners for both G strings and both D strings will be on the side of the headstock closest to you, while the tuners for both A strings and E strings will be on the side of the headstock closest to the floor, in order. As you tune, you generally want to tune in that clockwise pattern on the tuners, around the headstock, working your way down the instrument and higher in pitch. What makes tuning a mandolin more difficult than tuning a violin, of course, is that there are 8 strings instead of 4, meaning that you have to be precise or the instrument will be out of tune. It can be difficult to know which string is out of tune when you're striking them both in unison. Use "rest strokes" (in which you dampen each string with the pick after playing) to isolate each note individually as you're tuning. This will get a clearer tone on the electronic tuner, or whatever other tuning method you're using. As with any stringed instrument, you generally want to tune from flat to sharp, tuning the string up in pitch, rather than down from a higher note to the correct tone. This is because you want to settle the tension in the string toward the gear, not away from it. When you tune down, you risk letting the tension slip on the tuning gear as you play, making the string will go flat. This is especially true with new strings. Worn out or rusty strings will go out of the tune more easily and bedevil your fingers as you're learning. Make sure you change your strings out regularly to keep your instrument in tune. You don't need to change them every night unless you're Tim O'Brien, but consider changing them every 4-6 weeks of moderate to heavy usage. It can be frustrating to tune immediately after putting new strings on the mandolin, because it'll slip out after only a few minutes. After putting new strings on, each string puts lots of pounds per square inch of tension on the neck, and the wood will flex slightly. You need to account for this by getting the strings close and then letting the instrument rest for a second before fine-tuning. You'll get it in tune more quickly and more accurately this way.
Tune it like a violin. Locate the correct tuners for each string. Tune each string individually and both strings together. Tune up, don't tune down. Use fresh strings. Get the tuning in the ballpark, and then fine-tune it.