Sorry I won't be reading all the replies in this thread, but I hope I can still give you some sound advice on what to do. I wouldn't call myself amazing at fighting games, but I think I'm still above average and good enough that I enjoy these games.
Now from what I'm reading of the first page, I can see some of the posters saying that they practice in tutorial mode and think up strategies during the actual gameplay. I'm not quite like that, I have a very difficult time thinking in the middle of video games so I use my instincts instead.
So what I personally believe is important in these games is reaction time, if you can't react fast enough, the other players will hit you before your attacks land. Now that might seem like it's obvious, but judging from your OP I can see that you think other players have "priority" over your attacks. I've definitely have had that impression before, and it's usually because I wasn't fast enough to react to their own actions.
Now how would I suggest you improve your reaction time? Now that is a little difficult and I really wouldn't say I'm an expert at it. However, I think a good way of doing this is probably playing shooting games whenever you feel like it. Games like Call of Duty rely heavily on reaction time and I think once you get good enough at those games, it will definitely help you in fighting games (and probably train your eyes to occasionally pick up subtle movements in animation that help you determine what moves are coming next). Once you're able to pick up on these frames, you'll probably be able to block much sooner or even start using shorter framed attacks to counter them before it happens.
I also think you should probably pick someone who has a reach (hitbox area?) that you're very comfortable with. That's why I main characters with swords usually because their range is a little longer than other characters. You should familiarize yourself with the hitbox for each attack while playing other players (even if you don't manage to hit them) and from there you'll eventually start landing attacks. Now I say you'll familiarize yourself with the "hitbox", but it's not like I'm actually analyzing what the box looks like or how far it really is. I just use the attack enough times during the gameplay to start recognizing just how close the enemy has to be (and in what positions) for the attack to work. Once you become more familiar with the character, you'll probably start winning some games and enjoy the game more.
Now during actual combat, I'm not kidding when I say I rely on my instincts. I end up blanking my mind and I just use whatever buttons my fingers end up pressing. Now you're probably going to say that I button mash (and to be honest, I thought that at one point too); but I actually don't press the same button over and over and use different attacks and combos. Eventually you'll be comfortable playing most fighting games because you'll start to realize that there's an average sized hitbox that most characters use (like Ryu from Street Fighter). Also, you'll start to realize that there's commands which have similar functions throughout these fighting games.
I hope I'm not just repeating facts that you already know, but I really do feel like this is what I ended up learning throughout my years of playing fighting games with my friends (and alone). Good luck with your fighting games, send me a PM if you'd like to play some fighting games with me on a PlayStation console or the Wii U

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