First, Do No Harm

There are not a lot of ways that you can really screw things up with the Tube Amp Expander, but there are a couple and they're important.

Critically Important

These are the ones that can actually damage something (likely your tube amp) if you don't get them right.

• Make sure the impedance setting on the back panel of the TAE, in the section labeled "From Tube Amp" matches the impedance setting on your tube amp. Not getting this right can damage the output transformer on your guitar amp.^† And this is especially easy to forget if you're swapping the TAE between several different amps like I am.
  
• When powering up, turn the TAE on before the tube amp. When powering down, turn off the tube amp before the TAE. A powered-on TAE or an actual speaker should be connected to your tube amp any time the amp is turned on. Memorize this saying: "TAE is the first one to arrive and the last one to leave."
  

Important

These are the things you really should do, but it's not the end of the world if you occasionally forget. 

• Make sure everything is turned off when you're making connections. You should always be powered down when connecting loads to your tube amp, but there are some connections on the TAE that can be made while the unit is powered on (like connecting the foot controller, for example). However, the safest and easiest to remember is to just turn everything off when making connections. You'll never go wrong doing that.
• Make sure the TAE and your amp get plenty of ventilation. The TAE lets you work your amp really hard at low volume, and it does this by converting its power to heat. So you have to be really careful about temperature. Make sure you don't block the side vents on the TAE.
  
• Visually check your power tubes occasionally while playing and watch out for the plates glowing red. If you ever notice that they are, immediately power down. Then you'll need to replace the tubes. You may even need to take your amp to a tech if the new tubes red-plate as well.
  
• Check the temperature of the TAE and your amp every once in while by putting your hand on them (but not directly on the tubes!). They're both going to get warm, but if either gets substantially hotter than normal, power down and give them a break.
• I highly recommend unplugging all of your gear when you're not using it. It takes less than a minute and can save you a lot of grief. I once had a power surge take out 4 different pieces of home entertainment gear all in one go. Cost me a bit of money and a lot of downtime. I've heard of lightning strikes arcing across the circuitry built into surge protectors. Physically unplugging from the wall is the only surefire way to prevent AC spike damage. 

Critically Important, But Obvious 

These recommendations should be intuitively obvious, but have you ever seen somebody trying to drive a car that was obviously overheating or had a blown tire? People do incredibly dumb things all the time. And that's how serious damage or injury happens.

• Power down when an obvious malfunction occurs. If you see/smell smoke, if you spill something on your gear, if you're getting weird noises or sudden volume loss, all of these are billboard signs telling you that something is wrong. How you respond to that is the difference between a small repair bill and a really big one. Or maybe even a fire.
  
• Don't exceed the TAE's rated capacities. Duh. The tube amp can't be more than 150W. The speaker load you connect to the TAE needs to be able to handle 100W, or you need to keep the TAE's volume at a moderate level. The total speaker load also needs to be between 4 and 16 ohms. And don't use the wrong kind of cables; use actual speaker cables for speaker connections.
• Don't become an unintentional YouTube star. Don't make foolish MacGyver mods to your gear. Don't pile it up in unsafe towers of junk. Don't climb on it or put heavy things on it. Don't leave it in your car to be stolen or damaged from heat. Don't use 2-prong AC adapters. Don't play your electric guitar in the bathtub. Don't run 30 things off a single AC outlet.

 

† In reality, you can mismatch in one direction – using a "one higher" impedance setting on the TAE than on the tube amp. But I don't recommend it because it's easy to forget which direction is safe (using a lower impedance setting on the TAE than on the amp can blow the transformer). Some people like to deliberately mismatch impedance in the safe direction because it's less efficient (lower volume), but that's not really necessary when you're using an attenuator. I imagine there are some people who prefer the tone too, but it's a very subtle thing and not worth it in my opinion. I recommend just keeping it simple and ensuring that the TAE's impedance setting matches the tube amp.