The most important thing to know is that the gold in Klimt's paintings is gold leaf, rather than the iridescent paints are available today. The larger online art supply stores stock gold leaf (for instance Blick), while the Society of Gilders has a list of more specialist suppliers.
In Pip Seymour's The Artist's Handbook he's got a couple of pages on using gold leaf with tempera, but you can adapt the information from the techniques described by gilders and, especially, icon painters. American Artist magazine has a useful feature on Fred Wessel, who uses egg tempera and gold leaf "to achieve Renaissance luminosity". Between all these resources, you should have enough information to start using gold in your own paintings.
In answer to the question "Why not just use gold paint?", the Society of Gilders's FAQ says the reason is because "Gold paints are not really gold ... and will tarnish over time. Some of the newer paints supposedly will not tarnish, but even so, the resemblance to real gold is remote, at best. Despite the ease of applying this material, its visual and physical properties are inferior to those of gold leaf."
Personally, I'd to buy a tube of top artist's quality gold paint and a small gilding starter set, to see what each is like working with and how you feel about the results you get. Be willing to experiment, to create studies, rather than focusing on producing final paintings. There's nothing like trying an approach or technique of an artist you admire to give you another level of appreciation for their work.