I'm an amateur photographer looking for a wireless flash trigger. My gear is Canon 70D with Speedlight 380ex.
My question is: what should I look for in a flash trigger? How important is the trigger speed?
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Actually, the most problematic part of your equation here is the 380EX. It doesn't do Canon's wireless eTTL slave mode, which is probably why you're looking for triggers, but it has no manual power control setting capability, so even if you can get it to fire off-camera, you can't control the output and it can only fire at full power. And if you get TTL-capable radio triggers and use eTTL to control the power output of the flash, the problem is that the 380EX speaks e-TTL, not eTTL-II, so metering might be off.
You may want to consider swapping the 380EX for a newer eTTL-II flash (or 3rd party equivalent), that has manual power control and can be slaved to the pop-up flash in your 70D (e.g., a used 430EX/430EXII/580EX/580EXII--the IIs can also be camera menu-controlled)
In terms of flash triggers, it mostly depends on your budget and the method by which you want to trigger the flash. There are multiple methods to trigger. Physically, the triggering means are:
Each of these types of triggers also tends to break down into two subgroups: manual or TTL-capable. All the manual methods simply communicate the "fire" signal in sync with the camera so that the flash goes off while the exposure is being made. And that's pretty much all they do. They can't communicate anything else: TTL, HSS, remote power control--these all require the proprietary non-sync signals on the hotshoe, and the electronic communication protocol differs from brand to brand. So if you want all the goodies, make sure you're getting triggers compatible with your camera body's brand.
The speed of the triggers only really matters if you're going with a manual method, and will therefore be kept at or below your max. sync speed (in the case of the 70D, that'd be 1/250s). Propagation delays on the radio triggers can lower this even farther, but you're generally more worried about this with a full-frame camera, since the larger shutter travel can mean a slower sync speed than with a crop.
Length and robustness, of course. But primarily you'll be concerned with the connectors and whether or not your camera and/or flash have ports to hook the cable to. TTL cables will have full hotshoe-to-foot connectors with all the contacts/pins. Manual cables can use darn near anything that can communicate a simple short, but the most common connectors will be PC (Prontor-Compur) and 3.5mm (aka 1/8") miniphone. Cameras and flashes typically only have PC ports in them, but you can get hotshoe/foot adapters that can convert the sync signal to PC or 3.5mm.
Again, the connector or adapter comes into play in how to connect one of these to your flash. And some slaves can ignore a pre-flash which can be important if you plan to use the on-camera master flash in TTL. But Canon EX speedlights have a weird kick in their gallop and quite a few of the more noteworthy slave (e.g., Wein peanut) don't necessarily work well with the EX circuitry. The Sonia-brand green-based slaves, however, are known to work well, but cannot ignore a preflash. These slaves are manual only and must have line of sight to the mastering flash burst.
This is proprietary and can be more expensive than manual because you typically need both the flash and the camera to speak it, or you need another flash to be the master unit. It's also line of sight and range limited--especially outdoors in bright sunlight. But it's also going to be the fullest-featured system and the one covered under OEM warranty if you purchase OEM gear. There are 3rd party flash units, however, that can "speak" this protocol. And this is often the cheapest way for a newb to get started since with a new camera body and a new flash, they already have the gear.
So, you want to go with radio triggers, because you've read the Strobist, right? Man, there's a ton of different things to look for here, and they change nearly every three or four months, so understand that this is an area where the technology continues to evolve, and new features are getting introduced at a rapid pace. Keep up to date by following blogs like Flash Havoc or Lighting Rumors or the photographyonthe.net board's Flash and Studio Lighting forum. But here's a braindump of features to contemplate.