What would be the effects of galaxies colliding?

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For anyone who wants to see a cool simulation of a galaxy, merger, I can recommend a neat little applet): Galaxy Crash JavaLab.¹ Two more good sites with a bunch of different simulations are the GALMER website and this one (which turns out to have Samuel's simulation).

Cool, huh?

Anyway, let's look at some scientific work on galaxy collisions. Cox & Loeb (2008) simulated the future collision of the Milky Way and Andromeda. Here are some of their results:

There is a small chance that the Solar System will be flung out by tidal forces.
There is an equally small chance that the Solar System will be captured by Andromeda in the early stages of the collision, before the formation of the result of the merger.
Gas in the galaxies could be heated by an additional order of magnitude, from ~3$\times$10⁵ Kelvin to ~3$\times$10⁶ Kelvin. This alone could impact star-forming rates. Star formation is expected to increase anyway, due to other factors, although given that neither galaxy has a lot of gas, this will be minor compared with other collisions.
The result will most likely be an elliptical galaxy, leading to a different mass distribution and a slight change in the Solar System's orbit, even if most of the exotic tidal scenarios don't happen.

Van der Marel et al. (2012) brings up another good point: These two galaxies aren't the only players. The Triangulum Galaxy, M33, has interacted with Andromeda in the past and will do so in the future. This may be before, during, or after the Milky Way-Andromeda merger (which itself will take quite a long time). Both the Milky Way and Andromeda also have satellite galaxies, which could play important roles in determining the fates of a select few stars.

The addition of a third galaxy (which isn't as massive as the other two) has additional effects:

There is a 20% chance that the Solar System will go through M33 within 10 billion years, leading to a slightly higher chance of collision with other stars.
The Solar System will likely be pulled further away from the center of the result of the merger than it is now (although this may happen without the help of M33).

Finally, John Dubinski has written up an excellent overview (which is also quite non-technical) of the collision. He notes that a binary supermassive black hole will form during the merger, which could - depending on the precise path of the cores, eject the Solar System entirely.

Actually, I did find a cool simulation, and some images from it, each separated by approximately 170 million years: