Yeah, great idea!
We need something to go to Andromeda Galaxy
Well, I don't think the first spaceship should be able to go to another galaxy instantly. I was thinking about something amongst the lines of using Interstellar Drives(I admit, I am taking this from Spore(TM)).
This way, you don't need to make multiple spaceships, instead all you need to do is to add in new Interstellar Drives. They all have a limited parsec range. (Parsec is the space between stars, in case you don't know.) Here are all Interstellar Drives:
Drive 1: You can go up to 3 parsecs.
Drive 2: You can go up to 5 parsecs.
Drive 3: You can go up to 7 parsecs.
Drive 4: You can go up to 12 parsecs.
Drive 5: You can go up to 20 parsecs.
Drive 6: You can go up to 40 parsecs.
Drive 7: You can go up to 80 parsecs.
Drive 8: You can go up to 160 parsecs.
Drive 9: You can go up to 300 parsecs.
Drive 10: You can go up to 600 parsecs.
Afterwards comes the Intergalactical Drives. The first would allow you to fully explore your own galaxy, the next one will allow you to explore the sattelite galaxies that spin around the Milky Way, the third will let you go to Andromeda, etc.