The game (Railroad Tycoon II) loads and I can start a scenario, but it crashes every time I try to start a campaign. It seems most likely that this is because it's trying to play a video and can't, but I have no idea how to fix it.

Any ideas?

You're right, it does play a movie and it works for me. If it is a missing codec causing the crash, the only thing I have installed, as far as I know, is the Combined Community Codec Pack.

OS?
On Win7Pro x64 (Tested it now on a fresh win install on my Lenovo T410 8GB, Intel HD onboard GFX)...
The Videos (Smackvideo) are playing on my system - first start is a unviewable pixel mess...
after installing RT3 they where fine, but changing the windows theme "back" to Basic Aero again led to the pixaleted view... after installing ffdshow tryouts (32bit) it was ok again (even with just starting the exe and not the "link" with the compatibilities settings) until I again "activated" the basic aero theme (which really just means that I just choose the theme allready assigned)...
somehow there seem to be a problem with the needed fallback changes of the desktop-gfx mode, but beside that - the videos are playing (even if unviewable) on a fresh win7 install without any additional codecs (haven't looked for any the gog installer may have introduced as finding out about the installed 32bit codecs on a 64bit win7 is, without additional software, somewhat more complex than just opening the device manager)...
Just using a non-aero theme (Win7 Basic Theme) is avoiding the problem, but thats a nogo for me on systems that easily run aero (and i am just too used to the transparency effects nowadays)...

Edit: And MPC-HC just run's 'em fine out-of-the box, so the problem is most probably with the provided codec together with maybe a combination of desktop settings (as something RT3 changed whence starting seem to have helped with the RT2-pixelation) and gfx-driver (allways a good guess on such problems)... So I guess more tinkering with the compatibility settings (and be it with MickySofts ApplicationVerifier instead of the build in functions, something that next to Orca was an essential tool for every XPx64 User in not so old, old times) may lead to an acceptable result for the games playability

hey!

so I've had the same issue! I found out that many people switched from using an hdmi cable to vga and it worked for them...as I am playing on a laptop, that makes it quite impossible for me to do...so i wondered why that would be the issue...I think it might be the fact of switching from 32 to 16 bit color mode...I changed the color settings from 32 to 16 and it works now. If I am correct, vga automatically solves the problem as vga does not support 32bit color schemes...Hope that helps! It worked for me! :)

Yea, RT 2 is not working me, either. I'm getting a "Direct Draw #55" error that I only see if I tab out of RT2.

I found a direct draw fix and tried that. I got to the menu screen (NOT fullscreen) and it is all kinds of funky. I click "quit" and exit, but anything else is wonky as all get out.

Just reinstalled Windows 7x64 and am reloading my games. It has been a long, long time since I played RT2 last and don't recall if it worked when I got it here years ago.

I did try installing RT3, too, and that got me to the messed up menu screen. Before that, it was just a black screen with music playing.

UPDATE

I went into the properties in the exe file. It was set to Windows XP SP 2. I changed that to Windows 98/ME and now its working!

I hope this helps others :)

I can't post link there
Just google for - VEROK'S OFFICIAL BLOG RAILROAD TYCOON II (GL WRAPPER & PATCH)
It fixed for me problems with DirectDraw errors on campaign start. Wonderful resource for old games!!!!