Most companies that manufacture column kits for DNA and RNA extraction have a cleanup kit in their product lineup. These are really simple to use: you mix the DNA or RNA with a binding buffer, then put it through the column, then wash it once or twice and elute the purified product. I would highly recommend this option because it's fast and easy, and usually results in 80-90% return of your nucleic acid with very high purity.
If you have a lot of sample, an alternative option is to precipitate the nucleic acid with alcohol and salt. The procedure varies slightly for DNA/RNA: google 'DNA ethanol precipitation' or 'RNA ethanol precipitation' and you will find several protocols. If you do this, make sure to pipette your wash solutions (70% ethanol usually) over as many of the inner surfaces of the tube as you can, because if the contaminants are adsorbed on the plastic and they don't get washed away, they'll end up back in your sample. I would not recommend reprecipitation for low concentration samples, FFPE or small RNA.
NO! The protocol should be optimised for sample type and standard method should be used for all samples for reproducibility.
Recovery of samples will just mess up the whole sample and batch and thus samples would be scientifically useless. It can be done for the sake of getting "at least something" in the end but not a recommended practice.