I'm a bit confused about what's actually going on here because it sounds like you're dealing with two problems.
For the front derailleur, look down from the top and verify that the edge of the cage is parellel to the chainrings. Shift the chain to the largest cog in back and check that you can shift to each chainring. Shift chain to the smallest cog and repeat. If that's good then pedal and look for the point on the cage where the chain is rubbing. Turn you're cable barrel in or out 1/4 turn increments to alleviate the rub. If that doesn't work you may have to bend the cage slightly to resolve the problem.
For the rear derailleur, just make sure that you H and L limits are set according to the manufacturer and then check that you can shift up and down the cogs smoothly. If so you should be good to go. I don't know if it's normal for a chain to rub on adjacent cogs but if it's only slight contact I wouldn't worry about it.
Mike