There aren't really any effects that are only used on vocals. Hamonization and and pitch correction (AKA "Auto-Tune") are very rarely used on anything but vocals, but they are used in other situations for other instruments. All of the effects in your effect box could be used for other things besides vocals. The advice below also applies to guitar/bass/keyboard effects, and all effects in general.
How to learn to use a single effect
First, you'll want to learn the terms and parameters related to each effect and effect setting. The manual for the effect device in question is a good place to start, and there are many online resources beyond that. Be aware that some terms are very clearly defined and consistently used (e.g., pre-delay, mix, volume), and others are either commonly used incorrectly, have multiple meanings, or are hard to find clear defintions for (e.g., vibrato, tremolo, manual). Also, some settings may have very different names even though they are almost the exact same thing (e.g., hall, plate, and chamber are all types of reverb).
Second, listen for the effect in question in popular recordings. Listen to how it is applied and what sounds are created. Notice when a recording does not seem to have a popular effect on it. Look on YouTube for vocal-only tracks (many have been released recently, the vocal-only track for "Under Pressure" by David Bowie and Queen is an amazing demonstration of the use of reverb), and listen to how effects are applied to vocals in particular.
Finally, practice applying one effect at a time to your own voice using your equipment. Play around with extreme settings for each parameter of an effect, one parameter at a time, and work on a single effect with all the other effects bypassed or off. Get as much time as you can practicing with effects at the same volume that you would have at a show. At the same time, avoid taking up any band practice time tweaking your effects while your bandmates are standing around waiting. Instead, plan out and set up the effect(s) you want at home at low volumes, show up to band practice an hour early or so and tweak the effect(s) at high volume, and then listen to how the effect(s) sound(s) with the full band. Make notes of what doesn't work with the full band and tweak it after band practice or really quickly between songs. If the band members complain or something really doesn't sound right, just bypass the effects for that practice and try again next time, hopefully after fixing whatever wasn't working.
What order to learn the effects
Based on your particular unit, I recommend the following order, which should work in general with a few changes for anyone learning vocal effects.
- Play with the dbx AFS system at high volumes to get a feel for how it works. Turn off all the other effects when trying this out.
- Start with the Gold Channel turned On but all of the effects in the Gold Channel turned to zero or off. The EQ settings should have Low, Mid, and High set to 0 and MidFreq left at default.
- Turn on the Low Cut Filter and make sure it doesn't make you sound terrible. If it sounds all right, you probably want to leave it on.
- Learn EQ next. In general, cut (lower the EQ setting below 0) to make your voice sound better and boost (raise the EQ above 0) to make your voice sound different. Extreme boosts may cause technical issues, like feedback or un predictable compression behavior, and extreme cuts may make it hard to hear your vocals. Unless you're trying to get an extreme effect, avoid going outside of -3 dB to +3 dB. For the MidFreq control, either boost or cut the Mid control a lot (more than +/- 9 dB) and sweep the MidFreq back and forth until you find the sound you want, then bring the boost or cut back down to the +/- 3 dB range.
- The compressor is extremely simplified. Normally a studio or live compressor for vocals could have more than twenty controls, and rarely fewer than five. This one has a single control. Since you can't adjust it, I recommend using it very little or not at all. Perhaps try dialing it up to 20 - 40 and see if you feel like it's better. You can try cranking it up to 99 in a quiet setting to see what it does. Maybe it would be useful on stage for a certain song, but again it could be dangerous in terms of feedback and noise.
- Now the big one: Reverb. EQ and compression are more like tools to make the sound better and fit into the mix. Reverb is the first real effect that is almost always used to make things sound different, although very slight amounts of reverb can have a suble enhancement effect. I would pick a single Type to play with first, probably Hall, and then leave Tone alone, set Amount to about 50, and then play with Predelay and Tail. You probably never want Predelay at zero since that can make you sound farther away than the band. To sound like you are really close but in a big room, set Tail high and Predelay at maximum. After you have those two set, play around with Tone and finally adjust the Amount. You can spend your whole life learning about reverb and still not learn all there is. Effective use of reverb will be the single best result you can get from an effect box like this.
- Delay - If you have a good reverb sound, adding delay to it can make it bigger. You can also use delay without reverb for special effects and very short delays for interesting sounds. Delay is only slightly less popular than reverb.
- Modulation is at least as big and complex a world as reverb, but it's more optional. This is one you can use to get some really funny sounds. Certain settings might make it hard to hear you or make it impossible to understand the lyrics (but perhaps in a good way).
- Special effects that can be cool in the right situation: Distortion/Filter, Double, Warmth, Pitch FX.
- Effects that could be nice but are the hardest ones to get right - avoid these unless you really need them and have plenty of time to find the right settings: Harmony, Pitch Correction, De-Esser.
- Effects I don't understand why they are there and I might never turn on: Ambience, Noise Gate. You may find use for them, but I would put them at the bottom of your list to learn. The noise gate (and the de-esser, really) could easily cause more problems than they solve.
One important piece of advice from someone who has mixed many live shows and a few shows where singers have their own effects units:
Use the minimum number of effects. The more effects you use, the worse you will sound, the harder it will be to hear you, and the more dissatisfied the audience will be overall. Don't try to sound like a studio recording. If you have a sound person mixing your show and they ask you to not use it, I strongly suggest you trust them, don't argue, and don't use the unit. Explain to them the two or three most important effects and songs and give them a set list with notes on the same. Don't try to sound like a studio recording or expect the mixer to make it sound like that.