There are several ways you could go about identifying species through DNA. If you want to do everything yourself, the simplest option in terms of equipment needed consists of evaluating fragment lengths observed during gel electrophoresis after amplifying specific DNA sequences using PCR.
If you are content with some outsourcing, you can also send DNA samples to a commercial company for sequence analysis.
A compromise between these options in terms of information obtained, is to do study Restriction Fragment Length Polymorphism (RFLP) by amplifying DNA fragments and using restriction enzymes to cut the fragments, before analyzing the fragmentation pattern using gel electrophoresis. To perform RFLP analysis, you would need to obtain restriction enzymes in addition to the chemicals mentioned below, and they can be a bit pricey.
The minimum equipment would consist of a PCR machine, one or more pipettes with matching pipette tips, a gel electrophoresis tray with power supply and a transilluminator (preferably blue-light/non-UV). A centrifuge is not strictly necessary, but can be useful for processing/filtering your DNA source.
Some chemicals will also be needed: Polymerase and dNTPs for the PCR reaction (or a pre-made "master mix" containing both), electrophoresis-grade agarose and running buffer for the electrophoresis, along with a DNA dye specific to the type of transilluminator (Usually UV or blue light). For a blue-light transilluminator, GelGreen is a suitable DNA dye. You will also want to use a "loading dye" to mix in your DNA sample before applying it to the electrophoresis gel. This can either be purchased or prepared yourself by mixing sugar and food coloring in water.
You need some form of heating to dissolve the agarose - a microwave oven is convenient for this, but take care to avoid over-heating, glass explosions or flash boiling. It is convenient but not strictly necessary to have some lab glassware. Preferably use a screw-top bottle to mix your agarose solution. Always leave the top off when heating bottles.
Photographic equipment can be also useful for documenting results of gel electrophoresis.
Finally, you will need single-stranded DNA oligomers (primers) specific to the DNA regions you want to amplify. DNA primers can be bought from a number of companies, but it varies how easy it is for non-affiliated individuals to order and make payments. Macrogen has been my choice: They both deliver DNA primers and perform DNA sequencing.
You may be interested in the following thread on the DIY Bio e-mail group: https://groups.google.com/forum/#!topic/diybio/cPzfEuiZH58
I have collected some of the primer sequences mentioned in the thread on a page on OpenWetware: http://openwetware.org/wiki/User:Jarle_Pahr/Meat