This is a difficult question to answer with certainty because there is so much time between now and the milestones you describe. Colonizing Mars and space might happen in the next few hundred years (102 years). The sun will make the earth uninhabitable to life as we know it in approximately 600 million years (6 x 106 years), as it gets hotter and brighter. The sun will expand beyond Earth's orbit in 5 to 8 billion years (~5 x 109 years). The sun isn't large enough to explode as a supernova, but that doesn't mean it will be easier to survive it heating up and its eventually expansion.

These are very long periods of time, so long that it can be hard to imagine just how long that is. For context, life on Earth is approximately 4.4 billion years old (4.4 x 109 years), so we have at least that amount of time before Earth is consumed. Human beings have only been a distinct species for 6 x 105 years and civilized society is only 6 x 103 years old. Thus, we have two times as long as we have been a species (and 2000 times more than we've had society) to leave our solar system.

I believe it is reasonable to assume that we will survive long enough to solve this issue. Consider all the things we have accomplished as a species in the last 6000 years. Or even the last 100 years. However, I also think that it is very difficult for us to guess how to successfully leave this solar system at our current level of technology and scientific understanding. We will have to break the problem into much smaller ones and answer those first. Colonizing Mars is a good example - it is not strictly necessary to settle on Mars before exploring the rest of the galaxy, but we will learn many lessons in the process. Each step will bring us a little closer, just as going to space and the landing on the Moon brought us closer to Mars.

Thinking about "deep time" problems like this bring up a new question: when you ask "will we move to another galaxy, what do you mean by "we"? Humanity may change significantly in that amount of time - it might be hard for us to recognize ourselves then. We may also "survive" by sending information or biological programming out into the galaxy, rather than our human bodies. In your opinion, does this count?

If you are interested in answering this question, I recommend you look to science fiction too - scientist tend to focus on what do and do not know for certain (which is a lot), rather than what might be possible. You might start with a collection of short stories called "Year Million: Science at the Far Edge of Knowledge", Kim Stanley Robinson's "2312", Greg Bear's "City at the Edge of Time", or Ben Bova's "Mars" series.