So! You've just bought the latest Arcsys fighter on the block. Maybe you're excited to play as your favorite Persona characters; maybe you want to see for yourself what all the hubbub about this "Guilty Gear" is. Who knows? But in any case, you wind up on Dustloop because you're not a filthy casual and you want your game to have a little substance. That's good! You're on the right track.
But all these Dustloop members keep talking about doing stuff in Training Mode - mostly combos, but also some other things. That seems boring to you! You bought the game because you wanted to fight people, not beat up on a standing opponent for hours on end. Why should that be so important?
I had that exact thought when I first got into Guilty Gear. GG was the first fighter I tried to learn competitively, and I heard some pretty crazy-sounding things: "80% of your time with the game should be in training mode" was the most extreme advice I got. But I've come to learn that it's not that Training Mode is the most important part of a fighting game - after all, there are lots of things you can't practically train without a human opponent - but that unless your name is Tokido or something, it is crucial to your success as a competitive-minded player.
Why is Training Mode Important?
At any fighting game's most basic level, playing the game is about being able to consistently perform the action that you want. Sure, you can mash buttons and dash around like a newborn, but that won't win you very many matches after a while; eventually, you'll want to be able to do the right move at the right time in the right place. Anything else is essentially wild flailing, and wild flailing gets you killed.
The kinds of things you'll want to be able to do consistently include:
1) Poking (at what range is each attack optimal? Which moves are always good and which are situational?)
2) Movement (doing the right jump, the perfectly-timed airdash, etc. - in general, not being "out of position")
3) Combos (which combos work all the time? Which are situational? How can I best optimize my meter usage in combos?)
4) Pressure and hitconfirming out of pressure (how do I keep the opponent blocking? How do I mix them up? What do I do when I hit them?)
5) Reacting to and punishing your opponent's actions (my opponent likes to do -action-. How can I make him/her feel bad for doing that?)
While number 2 is a little bit difficult to practice without having someone to fight, numbers 1, 3, 4, and 5 can all be honed with the magic of Training Mode - and you don't even need another meatbag to do it with you!
The Basic Settings
The first time you visit Training Mode, there are a few settings you'll want to immediately adjust. Those settings are:
Information display: set the game to display as much information as possible. This will include damage, overall combo damage, and inputs, as well as maybe some other stats. These might be on by default, but if not make sure you fix that.
Ground recovery: turn on ground recovery. If you're playing Blazblue, don't make the opponent ground-roll, just set them to get up. You might want to mess around with this setting later for more situational practice, but this should be your default.
Air recovery: set air recovery to "back." I actually don't know how it works in P4A or BB, but in GG neutral air recovery is a couple of frames slower than backward or forward air recovery, so make sure your "opponent" is teching at the earliest possible time. I prefer setting this to "back" rather than "forward" because back recovery tends to put the training dummy in a good spot to keep doing whatever I was doing before. Again, you may want to change this for situational stuff (mostly tech traps) later.
Any other miscellaneous recovery settings: turn these on. In general, you'll want to pretend that your "opponent" is as good at getting out of combos as possible.
Meter: set this to max. You don't want to be dashing around trying to gain meter every time you want to practice a combo that involves a super.
Character-specific settings: check these out and turn on the ones you want. For example, when practicing Aigis in P4A, I find it useful to set "Orgia Mode" to "Fast recovery."
This is actually the simplest thing you can practice in Training Mode, but it is also the one you'll use the least. Any time I'm picking up a character for the first time, I like to pull them up in Training Mode and run through all of their normals and specials until I feel like I have a decent grasp on their ranges and what they do when they hit. I cannot emphasize this enough: since attacking is among the most basic actions one can perform in a fighter, it is critical to know what all of your attacks do.
But as I noted above, this kind of practice is actually pretty limited - a lot of being good at poking is match-up- and feel-dependent, so you'll get most of your practice poking when you're fighting real people.
Recommended settings for this kind of practice: Just the basic ones. If you're feeling adventurous, check out what all of your moves do on counter-hit as well - there will be a setting for counter hits that has options like "None," "First Hit," and "All." I recommend "First Hit."
Now we get to the meat of Training Mode. While combos are not, in fact, the most important part of playing a fighting game - no matter how good you are at combos, if you can't set them up properly you're still boned - it is nevertheless an incredibly good idea to practice them constantly, since they net you so much extra advantage. If this sounds overwhelming, don't worry! Most characters only have one to three "bread-and-butter," or B&B, combos that you'll have to rehearse. There are often many variants of B&Bs, but they're just that: slight changes in how you start the combo, different endings, etc.
The best way to practice combos is to just do them, over and over and over again. You'll probably mess up a lot, but don't get discouraged - eventually, they'll be so ingrained into your muscle memory that you won't even have to think about them. Every character subforum on Dustloop should have a list of introductory combos - start there and/or check out Challenge Mode if there's one available.
Recommended settings for this kind of practice: Go to the guard setting and set the opponent to guard after the first hit. This will keep you from dropping a link in the combo, then picking the combo back up again - you'll be able to tell when your timing needs to be tighter.
In the course of your exploration of your new fighting game, you might notice that sometimes your opponent blocks your attack. What happens next? When a character's attack is blocked, most of the time, that character can start some kind of pressure; in other words, the character can keep attacking the person who blocked in the hopes that they'll eventually mess up and eat an attack or throw. This is a critical part of basically every character's game plan, and it's just as important as learning combos.
Pressure strings (strings of attacks that force the opponent to keep blocking or take significant risks to stop) are like combos in that they often have some highly repetitive component from which the interesting parts of the pressure - the mix-ups - are deviations. For example, when I play Axl Low in Guilty Gear and put pressure on my opponent, it always involves doing some chain of normal attacks into Rensengeki, FRCing the Rensengeki, then running up and doing it all over again. This is what you should (at least initially) be practicing; if you're not sure what a good string is, ask around in the character's subforum or take a look at videos. Make sure you also ask about how to mix up the opponent within the pressure string, because otherwise, your opponent will just learn to block the pattern.
For more on mix-ups, check out Henaki's primer on them.
Recommended settings for this kind of practice: Set the opponent to always guard.
Hitconfirming, in fighting game jargon, is the act of realizing that you've successfully hit your opponent and beginning a combo from there (or, conversely, recognizing that you didn't hit your opponent and acting appropriately, either by starting pressure or by backing off). It should be clear by now why hitconfirming would be a good thing to know how to do: if you want to actually combo after a mix-up (or not get punished when your mix-up fails), can you really hope to rely on pure intuition and reaction time? Probably not.
There are a couple of different ways to practice hitconfirming. The first is more of a rote memorization kind of thing: whatever the point from which you want to hitconfirm into a combo is, record the opponent blocking incorrectly (for example, if you're trying to practice going into a combo off of an overhead, record the opponent blocking low for a number of seconds). The second is setting the opponent to block randomly and practicing recognizing when your opponent took the hit ("oh, now I combo!") and when they blocked ("keep pressuring!"). I find the second method useful more often, but the first one is good if you want to practice a specific mix-up.
Recommended settings for this kind of practice: I mostly covered this above. If you want to practice hitconfirming from counter hits, and I recommend you eventually do so, just set counter hits to "First Hit."
Practicing Reactions and Punishes
Unfortunately, you don't exactly have a monopoly on doing things. Opponents will try to pressure your character or mix them up, or you may just notice them doing the same action over and over. As the player on the other end, it's your job to react appropriately and make your opponent get off of you or stop what they're doing - hopefully inflicting some damage to him or her in the process. This is called "punishing" your opponent, for obvious reasons.
This one's straightforward to practice in theory but sometimes difficult to actually do, since you'll need to be able to record your opponent doing the action you want to punish, and for pressure strings that can sometimes be technically nontrivial. However, it's simple to practice punishing things like DPs or throws, and you shouldn't have any problem recording the training dummy doing those. EDIT: nstalkie points out that in some games, you may have the ability to record multiple things for your opponent to do, then have them play randomly. This is awesome practice for if, say, you're learning to react to an overhead - record two different pressure strings, one with the overhead and one without, and let the computer go to town.
Recommended settings for this kind of practice: Just record your opponent doing whatever you want to punish.