I'm having a hard time finding great, high-quality samples. Although I'm primarily looking for electronic, hip-hop, or house music samples your answer does not have to be genre specific.
How does one go about finding useful samples?
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For electronic, hip hop and house, you should be able to get a lot done with samples of the classic drum machines. Things like the Roland TR-808 and TR-909, the Linn Drum, and other classics. If you google around, you can find plenty of free or donationware sets. There are also commercial sets that are very nice, though most good software samplers will come with a set of classic drum machine samples.
Keep in mind that 'high quality' isn't always what you want. A lot of classic hip hop tracks were made with 12-bit samplers like the SP-1200 or the MPC60. Strategic use of bit reduction, compression, and EQ are critical to getting the right sound.
As you get better at turning raw samples into great samples, try to use synths to create new sounds of your own, then process those.
Donationware - be sure to kick these guys a few bucks if you use their sample sets:
If you want to make modern electronic music you will need something else that the good old TRs ;)
You can try a slightly different approach than pure samples if that's an option. As a producer of European electronic music I found ReFX's Nexus helpful from time to time. This is a so-called ROMpler (software based), a mix between sampler and synth, which basically you get the best of both worlds.
You will of course need a host/sequencer that support VST-plugins (ie. Cubase and many others). I would highly recommend you give it a try.
Here is a preview of what it can do (dub-step/electro-house, there are other demos of other genres):