I see our destiny as more of dispersing each to a different planet in the universe rather than moving in a tight cluster to a safer part of the jungle. All the same, if we are to discover a mechanism for travelling to another galaxy, we need to begin questioning our school teachers with some intensity.
Or just ask Larry Niven - he figured it out decades ago in "Bigger Than Worlds" - you simply [though simply does not mean easily] constrict the star at the center of your Ringworld into a propulsive jet and once you reach Bussard interstellar ramjet velocities the system becomes self-fueling from interstellar or even intergalactic hydrogen indefinitely...its only limitation is that it's STL but once you've accelerated the system to near the speed of light then time dilation moots that as a practical problem, at least for one way 'emigration' voyages.
Of course you need to be a Kardashev Type II/III civilization to be able to build one of those so it probably wont be THIS millenium, unless Megascale engineering is an unintended consequences of the Kurzweil Singularity if it comes, and presuming the Carter Catastrophe doesnt come first...
...but once you do build one...
A Flying Ringworld could conceivably carry the populations of millions of inhabited worlds [humans, posthumans, A.I.s, aliens et al] across the intergalactic gulfs on a Grand Tour of not only our Local Group but just about anywhere in this Universe and continue to do so for as long as our Universe lasts, even long after all the 'natural' stars succumb to heat death [if that is our Universe's ultimate Fate]...though if proton decay is real then eventually sometime before T+10^36 years even a Flying Ringworld would succumb to Final Entropy's icy grip - Thermodynamics 2 always wins in The End eh?
Furthermore I wouldnt want to disperse Humanity to PLANETS if dispersal WAS a good idea [and imo it is...right now Humanity's biggest danger is that all our eggs are in one planetary basket waiting for some dumb rock or human miscreant with a chimera virus to knock us off]...planets are 'stationary targets' for anything out there that might be 'bigger' than us and nastier than us [if THAT'S possible, lol] that are just too darn easy to find and *in Galaxy Quest alien voice* "hit em with a rock"...
Better imo to play the smart odds and AVOID settling planets - settle SPACE instead...
...with as many ships moving as far and as fast away as we can move em.
Scatter to the four winds and beyond ASAP and ALWAYS KEEP MOVING...if we stay away from planets as much as possible there's a LOT of 'empty' space to not be easily found in; SOME of us are likely to survive that way.
*in movie-Mexican voice*
Planets? Wee doan neeeeeeed noo steenkeen 'planets'...