Q:

Video preview plays at old frame rate after changing it

How I found this problem:

First, I opened Blender.
Then, I opened the "Video sequence editor."
Since I knew I would be working with videos at 60FPS, I changed the frame rate from 24 to 60 FPS in the sidebar.
Then, I dragged in my first video, and the audio and video lined up correctly (since I changed the frame rate).
I made a new view thingy to put the preview in by dragging from the bottom left corner of the screen, and I set it to "preview."
Then, I dragged the "end" slider at the bottom until it lined up with the end of the video, so I could see the whole thing.

Now, here's the problem. When I press play, the video preview plays back really slow (at 24FPS) while the audio is fine. Furthermore, when I pan the video to play back at a different point, the preview starts at the correct point, but plays back too slowly.
How can I fix this, so that the video plays at the correct speed?

A:

It seems that you were using a HD video source. AFAIK, Sequencer still works with single thread and not so much effective, relatively speaking. In that case, Proxy is born to well solve such issue.

Select the video strip you want to make proxy for, toggle Proxy/Timecode on the property sidebar;
Menu > Strip > Rebuild Proxy and Timecode Intices;
on Preview window sidebar, Switch the Proxy render size to the expected proxy percentage, which is just generated. As shown below:

Since the video quality would be significantly reduced, the playback speed is quite faster than the original source. However, VSE still uses original source for final rendering. 

Another way to keep the video/audio sync while playing preview is to use the suitable playback sync mode, which locates on Timeline header. (AV Sync or Frame Dropping)

Of course, you can use the above tricks all together to keep the sync best. However, if the playback is still slow or not sync enough, then you'd better turn to a higher performance PC for that. :)

