This is a tough question (scientifically)! Generally speaking, it's impossible to look at a DNA sequence and predict what function it might have. It's easier to go backwards: identify a function/trait of interest, figure out what proteins are necessary for that function, and then deduce the sequence.
I'll describe one strategy to go from protein to gene, and name some techniques which you can read up on if you want to learn more.
To find the amino acid sequence of a protein, you can chop it up using proteolytic enzymes. This gives you fragments of varying lengths, since the enzymes will cut semi-randomly at different sites in the protein. Run these fragments on chromatographic paper. You can deduce the amino acid sequence from the pattern, because larger fragments will move slower. This is the same idea behind Sanger sequencing (aka chain termination sequencing) of DNA, and was invented by the same guy.
If you know what the amino acid sequence is, you now know the mRNA sequence, since protein is translated from mRNA. However, in eukaryotes, you don't know the DNA sequence, since we have introns - i.e. bits of the sequence which get cut out between transcription and translation. We need a way to get from mRNA (protein precursor) to DNA (gene).
To do this, purify mRNA from a cell and generate a library of complementary DNA (cDNA) using reverse transcriptase. Then, sequence the cDNA library to generate expressed sequence tags (ESTs), which are subsets of the full cDNA sequence. Because the ESTs are subsets of the sequence instead of the full transcript, it's possible to "CTRL+F" the organism's genome and find the gene which contains the EST.
Scientists have been doing this for decades, so nowadays you can search a gene/protein database and instantly know which sequence is responsible for, say, a specific channel protein.
To go in the other direction from gene to protein to function/trait, you have to do a fair bit of legwork as well. In very, very simplistic terms, you "cut" some DNA, put it into a package, and deliver the package to your favorite organism. As you said, it's a bit more complicated than that. :)
For complex, polygenic traits, it's harder to (1) identify the genes responsible and (2) successfully engineer an organism to have all of those new genes. You're unlikely to see a human with tiger legs any time soon. It's much easier to give an organism a few genes that confer very specific traits. Look at this list of genetic modifications to crops and click on one of the traits; usually only 1-3 genes have been introduced.