A note before we begin, genetics is a massive field which ideally requires a degree to tinker with. No amount of information that can be provided on this wiki is equivalent to the amount of practice and time in the lab. This will serve as the basics.
All life runs on DNA, genetic engineering is the process by which we study the current genetic makeup of an organism, either by sequencing it's genome or various other techniques, and then modify that sequence in one way or another. This process is different depending on what you're hoping to modify. For example plants are orders of magnitude easier to modify than animals. This is because anything with moving parts has to be significantly more complicated to support itself and defend itself. All DNA is composed of 4 letters (it changes for RNA but ignore that for now) A, T, C and G. A always pairs with T and C always with G. Each piece of DNA is double stranded and one strand is the mirror of the other. If strand 1 is AATTGGCC then strand 2 would read TTAACCGG. However each strand runs in opposite directions. At one end of DNA is OH group that marks the end of the strand. This is known as the 3 prime end (3'). the other end, marked by a phosphate group is known as the 5 prime end (5'). When DNA is being synthesized it is Almost always synthesized in the 5' to 3' direction and as such it is said that DNA runs 5' to 3'. This is the bare basics. This does not cover replication or much else. For your reading pleasure I would suggest looking up DNA replication, Mitosis, Meiosis, leading strand, lagging strand, DNA polymerase (1 and 3), DNA transcription, DNA translation Protein synthesis, ribosomes, the basics of the cell, post translation modification of proteins, Regulation mechanisms of DNA (operators, enhancers, primers, sigma factors, etc). As you read this ought to point you in the direction of any other bits you're missing before continuing below.
Forget what the professor in second year taught you. Punnet squares are not what we need here. No breeding and backcrosses and test crosses and pedigrees. If you'd like to learn about all that, Youtube is your friend but it's useless to grinders.
For us sequencing is your friend. It is a group of processes that allow one to take an isolated piece of DNA and be presented with its exact chemical code. A note here, sequencing in all of it's forms is *expensive*. If you're lucky, the thing you want to work with has already been sequenced.
There are a large number of ways things can be sequenced, and the one you choose depends on how much money you feel like spending and the length of the bit of DNA you need sequenced.
Sanger sequencing (the old school method(read length 800bp))
It's great for short sequences and was the original method for sequencing. The process is fairly simple. Take your fragment of DNA, say some new gene you're working on. Run it through PCR (see below) so you have a whole bunch of it. Put a little bit in 4 small vials along with buffer and an RNA primer that matches the start of the sequence. Add in 1 type of dideoxy nucleotide to each vial (ddA in the first ddT in the second etc.) Then add all four of the regular nucleotides to all 4 vials. Finally add in DNA polymerase. What happens is DNA polymerase begins to synthesis a matching piece of DNA but when it uses a dideoxy nucleotide instead of the regular ones it gets stuck and stops leaving a short fragment. You then take each fo the four vials and load the contents of each into a separate well of an electrophoresis gel. Run the gel and the add a stain to show you the bands that form. By reading the bands from top to bottom you'll have your sequence. If this all seems to complicated it is because reading about it makes it sound like a nightmare. It's much simpler in video form.
So now that you know the bare basics of how this works, here is a link to the Wikipedia article detailing the other methods as there are now too many to list here without crowding this article up.
Getting something sequenced
Regardless of how they do it various companies offer sequencing for a price.
A fairly complete list can be found at Nucleics.