Yes, we currently have the technology and budget to build a moon base right now. This moon base could serve as a hub for reaching other planets, due to lower gravity and an extra push from its orbit. From there, we could go on to colonise many other planets in the solar system. It is possible to build things like ion drives that use little propellant, and thrust builds up over time, and physics tells us that it is possible to build warp drives, allowing us to travel faster than light. We do not have the technology to colonise other galaxies, or other stars, but we have potential, and in the next 1000 years, we could have control over the solar system, maybe with a Dyson sphere/swarm, and missions to other stars and maybe even galaxies, but for this to happen, we would have to set aside political conflicts that will not matter, and start thinking not years or decades, but in millennia. It is possible, but a lot would need to be done for it to be feasible.