It really depend how bad the scratch is, can you feel it with your figer or even finger nail? If not, then it's just minor scratches that can be remove with polish. If you're doing it by hand, I say use the Meguiar DACP (Dual Action Cleaner/Polish). If they don't do the job, go something stronger, but please be more careful when using more abrasive product as they might ruin your clear coat.
Here is an idea list of abrasive from most abrasive (10) to least (1): *Note* this is an basic idea, as the list may vary based on everyone's opinion. However, it'll give you an idea of what to go for if DACP don't work. Also, this list is taken directly from
www.autopia.org (valuable car detailing site), I just cut and paste to save you time.
10 MOST AGGRESSIVE - COMPOUNDS
3M SUPER DUTY RUBBING COMPOND HEAVY CUT 39004
MEGUIARS BODY SHOP PROFESSIONAL DIAMOND CUT COMPOUND 2.0 VERY HEAVY CUT #85
8
MEGUIARS BODY SHOP PROFESSIONAL COMPUND POWER CLEANER HEAVY CUT #84
3M PERFECT-IT III EXTRA CUT RUBBING COMPOUND 05936
MEGUAIRS MIRROR GLAZE HEAVY CUT CLEANER #4
7
3M IMPERIAL MICRO FINISHING COMPOUND MEDIUM CUT 39001
MEGUIARS MIRROR GLAZE MEDIUM-CUT CLEANER #1
3M PERFECT-IT III RUBBING COMPOUND 05933
6
MEGUIARS BODY SHOP PROFESSIONAL DUAL ACTION CLEANER POLISH MEDIUM CUT #83
5
3M PERFECT-IT II RUBBING COMPOUND FINE CUT 39002
MEGUIARS MIRROR GLAZE FINE-CUT CLEANER #2
3M ONE STEP CLEANER WAX, MEDIUM OXIDATION REMOVER 39066
MENZERNA INTENSIVE POLISH
3M PERFECT-IT SWIRL MARK REMOVER DARK CARS 39009
3M PERFECT-IT SWIRL MARK REMOVER LIGHT CARS 39109
MEGUIARS MEDALLION PAINT CLEANER #97
MEGUIARS SCRATCH-X #108
4 - POLISHES
MEGUIARS BODY SCRUB A10
MEGUIARS DEEP CRYSTAL PAINT CLEANER #30
MEGUIERS QUICK DETAILER #66
MEGUIERS SPEED GLAZE #80
BLACK FIRE DEEP GLOSS POLISH
3
MEGUIARS BODY SHOP PROFESSIONAL SWIRL FREE POLISH LIGHT CUT #82
MEGUIARS MIRROR GLAZE SWIRL REMOVER #9
3M FINESSE-IT II MACHINE POLISH 39003
3M FINESSE-IT II FINISHING MATERIAL 05928
MENZERNA FINAL POLISH
2 - GLAZES
MEGUIARS BODY SHOP PROFESSIONAL HAND POLISH VERY LIGHT CUT #81
MEGUIARS MIRROR GLAZE MACHINE GLAZE #3
3M PERFECT-IT III TRIZACT MACHINE GLAZE 05930
3M PERFECT-IT III MACHINE GLAZE 05937
3M IMPERIAL MACHINE GLAZE 05991
3M ONE STEP CLEANER WAX, LIGHT OXIDATION REMOVER 39006
1
MEGUIARS MIRROR GLAZE SHOW CAR GLAZE #7
3M PERFECT-IT III FINISHING GLAZE 05941
3M IMPERIAL HAND GLAZE 39007
BLACKFIRE ALL FINISH PAINT PROTECTION
Source:
http://www.autopia.org/forums/showthread.php?s=&threadid=18913&highlight=list+of+abrasive
If you feel like you need more help, feel free ask away or post on autopia.org. Autopia have many professional car detailers that can help you out. However, I believe they'll probably tell you the same thing.
In conclusion, the step will be:
1. Find a polish that suit your need
2. Wash your car and dry it
3. Try and find a shaded spot and work there.
4. Use the polish you bought and rub them in
overlapping circular motion. (overlapping as moving left to right, then up and down)
5. Use it until the polish almost dissappear, wipe it with micro fiber towel.
6. See if you can see the scratch. If you can, you will still see it after you wax the car. Thus, if you can, keep working on it several times. If it doesn't get rid of it, try to go with something more abrasive. (Becareful!)
7. If you need to work with stronger abrasive, repeat step 1-6 until you get rid of it.
8. Wax the car.
Hope this helps~ :-D