Time Machine is one of the best features of OS X, but it can be a complicated. And once you start having problems, and you make a mistake trying to fix it on your own, it can make things worse.
Senior Advisers at AppleCare who do nothing but work on Time Machine issues (and they are really smart people) have told me that you should never mess around with any documents or other data that you see in the backups folder via Finder (e.g., don't drag a document out of a folder in one of those backups). It can screw up a lot of things related to your backups and possibly corrupt them. The only way you should restore a document or anything from a Time Machine backup is by going to "Enter Time Machine," finding it in those backups, and and then clicking restore.
If that's not working for you, I suggest that you call AppleCare and get on the phone (not the Chat support) with the Time Machine/Wireless experts via this page. If you have purchased AppleCare, the support call will be free. If your AppleCare has expired (check to see if your support is active on this page), they may charge you $29.00 for a "pay-per-incident" support call. They can use screen-sharing to see what your issue is and figure out if there is a solution. If the solution you implement has any problems, the payment covers you for 30 days. To prevent losing your data on your hard drive (photos of your kids!) is worth $29.00. But since the Time Machine issue involves installation of new Apple software, they will probably give you an exception and not charge you. And the issue may not even be with Time Machine; it might be a problem with your external hard drive. Apple's experts can almost always determine that.
As for those of you who are experiencing long periods of time for your initial back up in Time Machine after installing Yosemite, this is obvious, but if you are doing it wirelessly, it's going to take longer, depending on how many documents, photos, videos, etc., that you have on your hard drive and the speed of your Wi-Fi.
The fastest way to do the back up is to turn off wi-fi and connect your your external hard drive to your Mac directly with a USB, FIreWire, or Thunderbolt cable. I've had bad luck with wireless backups, so now I only use a portable external hard drive that is connected to my iMac with a USB 3.0 cable all the time. The backups are more routine and more reliable. If you have a MacBook Pro or MacBook Air, you can still do the same thing by either by using a portable external hard drive that you can carry with you, or just connect it to your laptop when you get home and click on "Back Up Now." I use G-Drives and have never had a problem. They are reliable and already formatted for Time Machine.
Finally, after speaking with an Apple Time Machine Adviser, you may ultimately have to do a "clean install" of Yosemite and start over. Just downloading and installing Yosemite from the Mac App Store or using Migration Assistant, however, will just bring back whatever issue is causing you problems with Time Machine. It also doesn't really get rid of crap that builds up on your hard drive over time from previous version of OS X, such as old or corrupted .plists, caches, settings, and other things that you really need to wipe from your hard drive before installing Yosemite. A "clean install" we remove that garbage.
Here is an article on how to do a "clean install" but it involves using a "bootable install drive using a USB flash drive." Some people advocate this method, and I guess it works (most of the time). But this is really not necessary and it's far more complicated that it has to be (you have to use a Terminal command, which is something to avoid unless you really know what you are doing).
Check out comments section of that article and you'll see how people have problems going with "bootable drive." In the same comments section, I provide the steps for an easier, alternative way to do a clean install that works for me every time I do it. You can find the steps in the comment listed as "M.C. says 11/01/2014 at 6:10 p.m." This is the best way to wipe the crap off of your hard drive, install Yosemite, and restore via a Time Machine backup the only data that you really need to restore--your documents, photos, videos, and music. Your Applications (and even your music) can all be re-downloaded from the Mac App Store and iTunes (and they will be the latest versions). It's like having a new Mac.
Good luck.