Today I went to YouTube for the first time since I installed Fedora 19, and saw that I needed to install a missing Flash plugin in order to watch videos. I installed the plugin. Youtube videos then played fine, except no sound. I checked my sound settings and made sure everything was turned up/on, and tested both the left and right speakers using the troubleshooting tool. No sound from either. I've tried both the 'microphone' and 'internal microphone' options. Point me in the right direction?

If I understand you correctly the Youtube business is a red herring. Your problem has nothing to do with the flashplayer plugin. You don't have any sound at all, right? Have you tried running alsamixer from a terminal? You can make sure all the settings for your card are correct. What audio system are you using (alsa? pulse? other?)?
jdk

Right--no sound whatsoever. Now that I re-read my post, it wasn't necessary to include the bit about Flash and youtube at all.

I'll start googling alsamixer, and how to look up the type of audio system. I just needed someone to give me a lead. Thanks.

BTW, love your sig.

Glad to be of service and thanks for the kind comment. The alsamixer is found in a package called alsa-utils in Debian. Perhaps its package has a similar name in Fedora.
jdk

Looks like I'm stuck again. I've been using Google for a while, but I can't determine the source of the problem.

Here's the info for my sound card:

00:1b.0 Audio device: Intel Corporation NM10/ICH7 Family High Definition Audio Controller (rev 02)
Subsystem: ASUSTeK Computer Inc. Device 1297
Flags: bus master, fast devsel, latency 0, IRQ 43
Memory at feb3c000 (64-bit, non-prefetchable) [size=16K]
Capabilities: <access denied>
Kernel driver in use: snd_hda_intel

I have Alsamixer. When I go into Alsamixer and hit F6 for 'select sound card,' my choices are default, HDA Intel, and 'enter device name.' I've tested the speakers with both default and HDA Intel selected, but I still have no sound. Again, I've made sure all volume controls are turned up.

Also, here's this if it helps:

aplay -l
**** List of PLAYBACK Hardware Devices ****
card 0: Intel [HDA Intel], device 0: AD198x Analog [AD198x Analog]
Subdevices: 1/1
Subdevice #0: subdevice #0
card 0: Intel [HDA Intel], device 1: AD198x Digital [AD198x Digital]
Subdevices: 1/1
Subdevice #0: subdevice #0

Can you try this on a terminal
and put the path and name of any .wav file on your system. Maybe an interesting error message will appear. Here's what that looks like on my system--
and I hear a woman's voice saying "side left".
jdk

I tried that. Here's what I got:

aplay /home/dan/Downloads/hornytoads.wav
Playing WAVE '/home/dan/Downloads/hornytoads.wav' : Unsigned 8 bit, Rate 11000 Hz, Mono

Still no sound. It looks like Linux thinks it's playing the file without problems.

Make sure alsamixer levels are not muted, since alsa mutes them by default after installation.
Look for "MM" at the bottom of output channels and press m to toggle it (OO means it's unmuted).

It's working! It turned out that I just needed to go into the sound settings and switch from Analog Output/Amplifier to Analog Output/No Amplifier.

Well done! Persistence pays off, eh?
jdk

How does one access those settings? I'm having a similar problem, after having installed a new sound card.....

If you use pulseaudio, pavucontrol is the "control panel" you might be looking for, where you can select the output (cf. Profiles in the Configuration tab).