I can't help but feel stupid, but I'm running into the exact same problem on the Gog version as on my Ye Olde cd version of FS2. I can't get past the first training mission without skipping it.
I get to the point where I'm told to [press ] and maintain best speed. (Or highest speed, something like that). It won't advance or acknowledge that I'm doing this. It was the same all those years ago when I first got the game, and it's the same now.
I'd love to see how the rest of the training mission are like... Anyone know what I'm missing?

Well that's strange. Whilst I never played FS2 in the old days, the training mission you talk about gave me no problems in the GoG version. Could you do it again and record a video?
Fraps would be good for this, using the unregistered version only gives you 30 secs and has a watermark on it, but 30 secs should be enough to show us your problem.

It's a little picky on what you press... for example, if you have your speed set to a joystick gradual throttle, it won't pick it up sometimes. Make sure you know what button is set to be max throttle, because if its not default, it won't work. If you are using default buttons and pressing them then I don't know unless its some sort of key combo.

While in mission press "F2", or go into the options screen from the game lobby.
Check the cross like icon, that shows you your ship movement controls.
Find the item that says "Full speed" (or something along that line, I'm witting from memory over here). Select the item and press "Enter". Now press the key you want to assign to the function.
In game the new key should be shown in the training mission. If you have a non US-keyboard layout, the game will still show the US-layout "equivalent" of the key, but it should still be the key you mapped the function too.

I had a similar problem, but it was due to "pilot error!" hehe
I finally noticed that the little screen on the left of my HUD was indicating what keycombo to hit to activate the required commands! duh! LOL
I hope to find and excavate my old joystick (and then hope it works for FS2!) for this game!
edit...
Oh, i was flying with my mouse controls on Windows XP sp3
YMMV

Thank you all for making an effort to solve this, but these attempted solutions have all been... well, attempted previously. I've redone keybind settings more times than I can count (already back in the day when FS2 was new) and now as well. I've tried with and without joysticks, and everything else I can think of.
Still, it's no big deal, since I can just skip that one mission. It just bugs me as a gamer that I haven't really finished the game 100%.
Could anyone just tell me what happens after the speed increase? I assume it's some basic weapons stuff, which I of course know. It's nothing important at all, just kind of annoying that I couldn't figure it out nine years ago, nor figure it out today...

It's all basic stuff. Match speed, follow, stop, follow the guy around, fire guns, energy management, blow up a couple drones... that's about it.

Actually, I think it is annoying because if you dont finish the training missions you wont get your wings in the medal case (right in the middle of it).
Anyways, full throttle in FS2 is [backslash] , try finding that in your keyboard.
I'm assuming you've got regional keyboard settings like I do, so try also hitting some of the keys next to ENTER(or INTRO). Some keyboards even have a slim enter key and the backslash key on top.
But even though, you seem to be striving with something else, being that you've already mapped the full throttle key to another! I cant think of a solution to this one....

Thank you, but I doubt there's a solution for this. From what I can tell, after countless attempts at troubleshooting, the reason is so simple I should have known from the start. The universe is out to get me. Yep, it wants me to suffer. There can be no other explanation and no, I'm not paranoid.
Who told you that anyway?
There is no control problem. Trying two different joysticks and one joypad, not to mention a quite horrific number of different keyboard controls kind of rules that out. It is not a bug in the game since no one else has experienced this amount of difficulty with that mission. It's not something inherent to my installation of the game, since it was the exact same on my original CD installation on a completely unrelated computer.
The only common denominator would be the user, me. And since I'm fairly certain I'm not retarded, it's not my fault. Thus, it's the universe out to get me. There can be no... other... explanation...
Note to the universe: You can't break me. There's only one thing that could ruin me, and that's if you somehow ruin my Fallout 3 savegames. And Murphy, stay the hell out of my damned save folder!

Then lets go crazy with suggestions here
- Run it in Windows 98 compatibility mode
- Run it in safe mode (freespace's safe mode)
- Idem ^ (Windows this time)
- Check for cheat codes that give you completed objectives or something like that. Perhaps some kind of trainer?
- Make it run, chase your boxed edition with a shotgun, see if it reveals its true form
- Run away from it, go around the block and sneak behind it to scare the bugs outta the box
- Get an old CRT and punch it. leave your TFT intact, but look at it and say to it "See what happens??!? WHEN YOU #$$%&&#"!!!!!"
- Get a voodoo priestess, it wont help but they usually have few clothing items on them, so you should relax your tired eyes at least. Mind the snakes and curses though!
SOME of those have to help

1. Done. Didn't help. May or may not have unleashed the wrath of C'thulhu on the world though. Bonus!
2. Done. Made my left eye twitch. Didn't help.
3. *shudder*
4. Cheats of any kind are made entirely for the mouthbreathing demographic.
5. Don't have a shotgun anymore. Went over to Elk hunting.
6. No blocks around here. Running around my house would just look silly.
7. My old CRT is needed for my 486. No dice.
8. Already promised the Underdwellers my first-born. No result yet, but the shadows move in my house every time I notice impossible geometry. Yesterday I saw two parallell lines intersect. Could work in time...

Is this the same computer from long ago? (Please tell us about it)
If so, are there any special settings we should know about (special keyboard, etc)?
What OS are you running it under?
Are you using Freespace 2 Open? (if you are try running the debug executable)
Have you tried deleting the pilot and starting a new one?
I have also had this same problem but I never solved it. Sometimes I could get through the mission and sometimes I couldn't.

Hmm make real sure that you actually reach the default top speed the craft can attain with energy (engines, gun and shields) equalized. No idea if it works, I'm drawing on memories of script heavy games such as the CoD series where if something wasn't done like it wanted it done, you were stuck.