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Greg Egan's Diaspora sets out how humanity could explore the galaxy and even the multiverse, which if you can't be bothered reading consists of:
In the here and now the only way to travel to another system with our current tech is via nuclear pulse engines.
Basically you build a large spaceship. Stick it on massive shock absorbers which are in turn connected to a metre plus thick steel plate.
Cut small hole in the middle. Have a door that opens closes.
Eject 1kt explosive device out door. Repeat 500x till you get to orbit.
Basically you could get a spaceship up to very high speed with nuclear pulse engines to turn a multi hundred year journey into less then 100 years.
That said the biggest problem with interstellar journeys is that our material science and manufacturing tolerances are pretty shit. Essentially all of the air will leak out through the metal skin of the spaceship.
I still think carving put an asteroid, sticking engine on it (see nuclear pulse engines) , covering it in ice and water will solve the problems radiation shielding, losing critical gases and provide ample fuel and water for a very long journey.