Goth music - much like caviar - can often be an acquired taste. If you haven't previously listened to any music from this genre, it may be very different than what you are used to, or even from what you are expecting. If you're looking for a blaring wall of electric guitars and lots of thrashing, thumping, headbang-worthy noise, be prepared for something utterly different.
Gothic metal bands, such as: Within Temptation; Nightwish; Leaves' Eyes; Epica; Sirenia; Xandria; Tristania; After Forever. These are, obviously, more metal-based than Goth music, and tend to have operatic feminine vocals, occasionally with a deep, growling male voice as counterpart. They have a very dark and beautiful atmosphere without many of the dirgelike qualities of genres such as post-punk.
New Romantic and dark pop musicians include Adam and the Ants; Kate Bush; Tori Amos; Stevie Nicks. (Yes, I know I'm lumping two genres of music together here - be grateful, I considered sticking David Bowie into my mash-up there as well. Yikes - genre clash!) Adam and the Ants and Stevie Nicks in particular have been credited with guiding at least several Goths 'of a certain age' into the Goth subculture to begin with. These musicians are slightly more unconventional in sound than the Gothic metal bands above.
If you listen to punk bands such as The Damned (also very popular amongst Goths - in fact there is some debate over which genre they currently belong to), The Sex Pistols, The Clash, and The Boomtown Rats, then you probably have a pretty good idea of what early Goth rock/post-punk bands such as Siouxsie and the Banshees sound like.
With all of these kinds of bands, I'd recommend that you find songs other than their singles, to really give you a feel for the music. Note: with all of the bands listed above, as well as with Goth bands, if you don't like it, stop listening. There is no point in 'training' or forcing yourself to like a band or genre that you clearly do not like. I would, however, like to point out that, because of the dozens of subgenres within Goth, you may wish to try listening to a different 'type' (more on this in just a moment).
When you want to start checking out albums from Goth bands, good starting points would be highly influential bands such as Bauhaus (I haven't started wibbling on about how Bauhaus pretty much defined the Gothic music scene yet? How remiss of me!) - try their infamous track Bela Lugosi's Dead, credited as being the song that started the subculture - Siouxsie and the Banshees, The Sisters of Mercy or Christian Death.
Later, you may want to try more specific styles of Gothic music, for example:
If you like electronic music, I'd recommend looking into EBM, Industrial, darkwave and dark electro artists such as: :wumpscut:, The Cruxshadows, Das Ich, Einsturzende Neubauten, Unter Null, Dulce Liquido, VNV Nation and Collide.
If you're into musical theatre, you may enjoy dark cabaret or punk cabaret such as: Jill Tracy, Voltaire, Rozz Williams, Katzenjammer Kabarett, The Dresden Dolls, Vermillion Lies, Stolen Babies, Emilie Autumn, Hannah Fury and Rasputina (some of these bands are considered Gothic, some of them are closely related. None of them will cost you any Goth points, anyway...).
If you prefer softer music, for example ambient, you may prefer dark ambient, ethereal and neo-classical bands such as: Rising Shadows, Black Tape For A Blue Girl, Dead Can Dance, Lycia, and Autumn's Gray Solace.
If you prefer music with a New Age or Celtic feel, try listening to bands like Faith and the Muse, FAUN, Dandelion Wine, Inkubus Sukkubus and Corvus Corax.
The lyrics in Gothic music are often very deep or meaningful - they can be wistful and poetic, sharp, ironic, witty, or referring to myths and legends, often Grecian or Pagan. Subjects that are rarely covered are shaking one's booty, being taken to the candy shop, dancing in da club, drinking copious amounts of alcohol or smoking illegal substances; so listening to the lyrics can actually be enjoyable rather than making you cringe. You can search online for lyrics if it's difficult to understand what's being sung.
Remember, Goth music isn't Satanic or scary. Like all music genres, it has its own flavour (or should that be 'flava'?). Also, do keep an open mind to all types of music - don't reject something just because it isn't Goth. You wouldn't believe the amount of Goths I know who are into Lady GaGa (myself included...).
Listening to: Bibit Aleum - Corvus Corax