Major problems with fonts after upgrading to Yosemite

You likely have a lot of garbage fonts installed. Follow these steps to get your Mac back to only the fonts installed by OS X.

1) Open Font Book and choose Restore Standard Fonts from the menu. All third party fonts will be moved out of the System and root Library Fonts folders to a new folder next to each one named Fonts (Removed).

2) Click on the desktop so Finder is the name of the foreground app next to the Apple logo at the upper left. Hold down the Option key and choose Go > Library. This will open the Library folder in your user account. Create a new empty folder on the desktop. Open the Fonts folder within your user account. If there's anything in it, move them to the new folder on the desktop.

3) Restart in Safe Mode by restarting and holding down the Shift key. When you reach the desktop (you'll need to login in Safe Mode), restart normally. Among other things, this clears all cache files from the user account logged into in Safe Mode, and resets Font Book's database for that account.

4) Clear all remaining font cache files from the system. Close all running applications. From an administrator account, open the Terminal app and enter the following command. You can also copy/paste it from here into the Terminal window:

sudo atsutil databases -remove

Terminal will then ask for your admin password. As you type, it will not show anything, so be sure to enter it correctly.

This command removes all font cache files. Both for the system and the current logged in user account. After running the command, close Terminal and immediately restart your Mac.

Now test. With only OS supplied fonts available and all font cache data and other such settings cleared and reset, fonts should display properly throughout the system.

There must still be non standard fonts loading, or you'd never see the weird font pictured above when bolding text in Gmail. That isn't anything supplied with OS X.

If you did run Restore Standard Fonts from within Font Book, then they must still be in your user account. Did you move any existing fonts in your user account to a new desktop folder, or just copy them?

Check again. (A different way to your home folder) From the desktop so Finder is the active app, press Command+Shift+H. This will open your home account. If you don't see the Library folder, press Command+J. Turn on the check box to show the Library folder. Close the dialogue box. Now with the Library folder showing in your user account, double click Library, then the Fonts folder. Is it empty, or are there still items in it?

Well, this is quite bizarre, then. With only OS X supplied fonts left in the System and root Library folders, and your user account Fonts folder empty, all of these issues should be gone.

Are you using another font manager such as Suitcase or FontExplorer X Pro? The steps above should reset Font Book so third party fonts are no longer open. But then, Apple recently changed the way Font Book works - again. The database it wrote used to keep track of what fonts were activated. Now it keeps track of what has been deactivated. At least in El Capitan it does.

If you have no other font managers installed, launch Font Book. A reset as you stepped through above won't change anything in FB if you added fonts as a library set. Those activate fonts in place (from whatever drive they reside on) rather than specifically being in a Fonts folder. So those may still be being activated at each launch. Go through all sets listed in Font Book and disable anything that's active. Test again.

If no help there, go into the System Preferences and create a new test user account. Login to the new account. Test Gmail and other areas where you were seeing issues. If all works as expected in the new account, then the problem is somewhere in your normal user account. If it behaves the same, then it's a system wide issue.

Are you sure that this is also a font-related problem?

Well, we know it at least partially is/was. The steps you followed fixed the system dialogue boxes. That artsy font shown in your second image sure the heck shouldn't be showing up simply by choosing bold text when composing an email. That's normally a sign of a bad font that has incorrect internal name. Such as the Brady Bunch font (not the newer Brady Bunch Remastered font). Here's the problem:

Notice the incorrect full name. When activated, that causes it to conflict with the real Arial font. After activating the font, just about anywhere Arial regular should be being displayed will suddenly change to Brady Bunch. That's what I'm presuming is happening with that strange "splatter" font, but can't be certain without knowing what the name of the font is and where you got it from so I can look it over. It's why I had you remove all third party fonts from the system, or at least disable them. That should make all such issues go away.

We now know with your last post that this is only happening in Chrome. So have you tried clearing all cache data in Chrome's settings? Instructions to do that are here. It may not help as it would be rather unusual for a browser to store font cache data.

Excellent! Learned something new. Chrome apparently saves its own font cache data.