Is this normal or am I doing something wrong? I get the intro video but seemingly non of the in game cut scenes. Can someone point out where they should be so I can confirm if they are there?
This is occuring in both games for me.

eh? you're not expecting cutscenes after each mission are you? :)
apart from intro/outro I recall 3 cutscenes from fs1, the moon scene before first encounter with shivan, boarding shivan vessel and discovering shield technology, tiny animations during briefings is all you get most of the time :)

OH I know that, I know exactly where the shivan shield cutscene is cause I played the crap out of this game in college, It went right past and never played it. I will do some digging though, its entirely possible I messed with something.

I also appear to be missing cutscenes. When I try to play Introduction in the tech room, the screen goes black for a second, and then it goes back to the tech room.
INTRO.MVE (195 MB (205,252,206 bytes)) is located in the data2 folder inside of my FreeSpace2 folder.
I have previously installed FreeSpace Open on this computer, and I installed to C:GamesFreeSpace2 instead of the default location.

I have the same problem. The movie files are there and apperantly I alreayd passed several points in the plot that would give me a movie sequence (Including the intro) however I get none.
When I try to play them in the movie room nothing happens simply. A solution would be appreciated.

Are you playing the OpenFS version, by any chance? Try tossing the MVE files into the dataMovies folder.

I have the same problem with both games. I'm gonna try Quinch's solution and post later if it worked.
To post some spec's that might help to find the cause. I am playing with the latest stable fs2open, with a 8800 gt, and in 1920*1200 resolution.

I've searched around and found what I was doing wrong: when using fs2open I forgot to select mediavps as my mod in the MOD tab of the launcher.
Don't know if this helps anyone else (since there is another post in the GOG with clear instructions to install fs2open: http://www.gog.com/en/forum/Freespace_Series/gog_freespace_2_freespaceopen/11), but I'll go ahead and post it just in case anyone had the same problem.

The cutscenes should work with the retail or FS2Open exe, with or without using the MediaVPs as a mod. If they don't something is wrong.

I found that they werent loading from the proper location when using FSO. I ended up downloading the ogg versions of them with the loader and that fixed the problem.
Now I just need to get the lucifer scanned without dying.

Because of issues, the FSO wasn't given the code to play movies. So they coded the abilty to play in .ogg format