As far as colonizing the galaxy goes. I don't have the exact numbers of the milky way in my head anymore, but wouldn't it be possible for humans to just go from planet to planet slowly if we were capable of moving around close to the speed of light?
You can't just jump to the speed of light. You need to
accelerate large amounts of mass to 99% the speed of light - it takes INCREDIBLE amounts of energy (E=MC^2 and we know what a large number C is).
Currently, the slowest form of propulsion, and
the most fuel-efficient, is the ion engine. For the first half of your journey you would be pressed into your seat by acceleration, then the second half you would be turned around and pressed into your seat by deceleration. I guess we would probably need to be unconscious.
Proxima Centauri is the nearest star, and it is ONLY 4.24 light years away.
So if ionic propulsion were to be used for a mission to Proxima Centauri, the thrusters would need a huge source of energy production (i.e. nuclear power) and a large quantity of propellant (although still less than conventional rockets). But based on the assumption that a supply of 81.5 kg of xenon propellant translates into a maximum velocity of 56,000 km/hr (and that there are no other forms of propulsion available, such as a gravitational slingshot to accelerate it further), some calculations can be made.
In short, at a maximum velocity of 56,000 km/h, you would take over 81,000 years to traverse the 4.24 light years between Earth and Proxima Centauri. To put that time-scale into perspective, that would be over 2,700 human generations.
Our Galaxy is 980,000 light years across. :/
That would mean something like 2 million years at 50% the speed of light to get across. Let's be generous and overestimate to upwards of 10 million years to colonize whatever can be. That's virtually nothing compared to the time until a potential heat death, which may or may not even take place.
Without any means for teleportation/breaking the speed limit it would certainly require many generations of people to even get to every place that humans can settle in, but it's not impossible. And if humans don't get wiped out by themselves or some natural extinction kind of event, I don't see why we wouldn't end up colonizing the galaxy. In fact, it would seem inevitable to me if humans don't die out. Who knows if whatever reaches different places can still be called human though, the differences are bound to be quite drastic on those time scales. Communication would be pretty much nonexistant without superluminal transmission of information as well and a holiday trip across the galaxy won't be possible either, which makes the whole prospect at least as exciting as sobering.
As far as energy is concerned:
https://ourworldindata.org/wp-content/uploads/2014/03/global-primary-energy-consumption-1800-2015-1.pngAnd we're still absolutely terrible at harnessing the energy output of the sun. Plus, if "uploading human consciousness" turns out to be possible, the amount of mass that would have to be sent through space could be drastically lowered. That would also allow for conscious travel, perhaps within a digital world. Although that would seem to make the whole endeavour redundant beyond climbing a mountain simply because it's there.