These achievements should unlock during normal gameplay.

Play the first three Quick Race game modes to get used to the game mechanics and unlock Speed King, Survivor, and Eliminator on easy difficulty.

While playing Tournament mode, try to not use any weapons for one race and unlock Pacifist.

While playing through Tournament mode, Ace Pilot, Kamikaze, Defender, Tractor Factor, Bomb Disposal, and Proximity should all unlock serendipitously.

These achievements may also unlock during normal gameplay, but there is also an easier way...

From Quick Race, choose the Survivor game mode > Infra tournament > Dreksu track.

At the starting line of the track, there are a bunch of turbo pads and item boxes. You can hide around the corner of the fork directly above the item boxes as shown in this screen shot:

As the CPU racers whittle away at each other's health while racing around the track, you can grab weapons from this vantage point. You can shoot or drop bombs and mines from this safe spot to unlock the achievements. Keep grabbing and using up weapons while the CPU is moving around the track until you get the weapon you desire.

Unlock all ships

The description for this achievement is wrong. It comes from an earlier version of the game. There are no ships to unlock in this game. Simply use each ship once to unlock the achievement.

Unlock all tournaments (Hard)

The description for this achievement is wrong. It comes from an earlier version of the game. There are no tournaments to unlock in this game. I'm not 100% sure about the parameters, but I believe you just need to earn a medal in each of the Easy tournaments. The achievement unlocked for me when I earned special medals in the Easy tournaments.

To earn a special medal you need to come in first place on each of the four tracks within a tournament.

Special medal for Super tournament (Hard)

Earn a special medal on any of the Hard tournaments.

Earn all the Special medals on all levels

This is the most difficult achievement in the game, but it is completely within reason. I have three bits of advice that can help you out:

1. The speed of the ships increases by 10% to 15% as you increase the difficulty level from Easy to Medium to Hard. Because Easy difficulty is the slowest, it helps to raise the number of laps from 3 to 5. You need more laps to move into first place and build up a comfortable lead. On Medium and Hard difficulties, fewer laps are better. The faster speeds make it much more difficult to maintain the lead once you've taken it.

2. You will need to use different ships on each tournament and difficulty level. Each tournament's tracks favor speed (if there are a lot of straightaways), or acceleration and steering (if there are a lot of sharp corners). The way the tracks play changes yet again with the increased speed of higher difficulty levels. You'll have to experiment to determine which ships work best for you on each tournament. If you keep playing one tournament over and over to no avail (say more than an hour), try another ship. For myself, I can only say that I mostly used a combination of Ranger and Sting on Easy and Medium, and Roza and Krait on Medium and Hard.

3. Use Restart while in a Tournament. If you are playing the second, third, or fourth track in a tournament, hit Esc (pause) right before you die or the CPU crosses the line in first place. Press Restart to start that track over again while retaining first place in all of the previous tracks. You can do this over and over again until you earn the special medal for the tournament.