WARNING: WALL OF TEXT OH GOD IM SRY
The neutral game is something I struggled with for years and had a difficult time grasping the concept.
Neutral game tutorial (gold gold gold, juicebox is king): https://www.youtube.com/watch?v=FQQCan5oo90
Footsies scenarios (really useful, but only if you grasp the concepts in Juicebox's video / that I'm about to describe): http://sonichurricane.com/?page_id=1702
A simple explanation on countering projectiles: https://www.youtube.com/watch?v=7ws7YLLQBPU
Playing the neutral game rests entirely on you predicting what your opponent will do and countering that accordingly. For example, I'm a Sakura player. Let's say I'm playing versus you, and you know that I'm likely to walk forward and try to hit you with my standing heavy kick. In ANTICIPATION (not reaction - as in, you conciously went to yourself "I think this Sakura will do this action in a moment, so I will pre-emptively act to counter that action") to this scenario, you want to defeat my attack. As described in the first video I linked, you can do this in 2 ways:
You can beat all moves (with some wonky exceptions) with the 2 above methods: Hit the opponent's attack before it's active, or make the attack miss and hit the it's recovery. You can defeat MANY players by just doing the above as they won't adapt to you beating their attacks.
Here's where the full neutral game comes in: A good player will adapt to the 2 actions above. They will, based on your previous counters to their move, force the same situation and counter your counter. For example, let's say in the Sakura Standing Heavy Kick scenario, you walked forward and hit the startup of my attack with yours (i.e. used method 2 listed above to counter me). The same situation arises again. Knowing that you went to option 2 to beat my heavy kick, I instead DON'T walk all the way forward, stand still, and press nothing, letting you whiff your move that was to counter my kick, and hit your recovery. As in, /I/ use option 2 to counter your counter - by making your attack miss.
The situation arises again. From your point of view, you want to counter my counter using either method. Then I want to counter your counter. And so on.
To practice this in your own time, go to training mode and think of a time vs a character you were completely dumbfounded on how to approach in neutral. Let's say its Abel vs Bison. Record Bison (as the training dummy) doing exactly what that opponent did that you couldn't deal with. Use the above steps listed to find out how to punish the actions. Once you know how to counter each of your opponent's options, it's a matter of deciding in a match what your opponent is likely to do in the next moment, and positioning yourself to counter that.
To gather info on what your opponent is likely to do mid-match, consciously force situations and remember what your opponent did. E.g. I'm Abel, vs an M.Bison. Every time I walk into his LK Scissors range, he throws it out immediately at me. So, knowing this, I'm ready to counter his move, using option 2. I ALMOST walk into the range of his LK Scissors, walk back a little, and wait for him to do the attack, and hit his recovery if it happened.
Let's say the M.Bison catches on to this, he knows I'm baiting. So he doesn't throw out any LK Scissors. This Bison is happy to just down-back and not act. (this is the part I couldn't figure out for a long time). You counter this by walking at him. Just hold forward. If he REFUSES to act, you get up into his face and throw him. He has to attack to keep you away. If he does attack to keep you away, now he's attacking again and leaving himself open to options 1 and 2. See where I'm going with this?
Summary:
Sorry for the longwinded, scattershot explanation, but if it helps one person I'll be happy. If anyone has further questions I'm happy to help via PM.