Nuclear Pulse Propulsion can get us to the stars at a maximum velocity of 0.1C assuming that you have to decelerate again, if not you can reach 0.15C which will be important for latter. Also we will assume that there is no faster than light travel. Because that would open up another can of beans since humanity could convert an amount of mass equal to the observable universe into human flesh in 6,000 years if the 1994 population growth rate could be maintained. And it could be maintained if the superluminal travel would allow humans to access new resources fast enough.
So assuming that you could send out self replicating probes that would travel at an average velocity of 0.1C and that it would take each probe 100 years to make a 100 copies of itself each time that they entered a new stellar system, you could still visit every stellar in the Milky Way in under 1 million years. If theses probes carried humans or mechanized human descendants then our civilization could also spread as quickly.
If you programmed these probes to seek out life bearing exoplanets and crash into them an top speed, which for nuclear pulse propulsion is .15C if you burn all the fuel and don't plan on decelerating, you could purge all the life bearing planets in the galaxy and keep yourself safe from aliens forever. And if you programmed those same probes to accelerate toward any repeating, terminating radio signal and crash into the source body you could all target intelligent species. For that matter you could have them detonate there bomb stockpile before impact to scatter relativistic shrapnel cones at space habitats.
And that doesn't even account for the danger of other species colonizing everything and taking all your resources. Basically your only real option to keep yourself safe is to attack with berserker probes first and hope that you can get everybody before they launch their own. So you need at least a million year lead on your galactic war to ensure your own survival.
In a universe were superluminal travel cannot exist, any species that creates even the most primitive form of relativistic vehicle and self replicating machinery is an immediate and lethal threat to all other forms of life. Because of the rate at which intelligent life can spread, you pretty much have to attack all alien intelligences without provocation and with even knowing if there is anyone to attack. As previously stated probes with a maximum practical travel velocity of .1 C could visit every stellar system in the galaxy in as little as 250,000-1,000,000 years.
Now with speculative but still possible technology it could be done even faster. Even if we discount confined fusion as an energy medium for a reaction engine, hydrogen/antihydrogen reactions could still be used and we know they would work if there was enough antihydrogen. The solution to that being solar powered particle accelerators orbiting in the inner solar system. They would take in hydrogen from the solar wind and produce antihydrogen using abundant energy. A laser assisted launch could also increase the acceleration of outgoing ships of any type if a large solar powered array of lasers with an output greater than a terawatt was used.
All this culminates in the ultimate plan for species survival, to completely disassemble the all the rocky planets of every stellar system we can get access to and use that mass to build Dyson Swarms of space habitats around the local stars. That way we can continue to survive around those stars until they burn out, which in the case of the lowest mass red dwarfs would be about 120 trillion years. The largest lofstrom loops possible with current engineering can lift 500 million tons a year and since you could only fit about 1000 on earth it would take nearly 10 million years to disassemble the planet. But it can be done only with proven technology, no super materials or new energy sources needed. You could power them using huge convection towers that contain liquid halite, which would be heated by the hot lithosphere you are uncovering. And of course the job would only get easier as the planet is taken apart: less gravity, more heat being radiated, more materials for building and maintaining the loops. That said you still have to use nuclear pulse propulsion to move the material for the first loop into orbit, about 2 million tons of it. But with nuclear pulse propulsion that is doable. We can conquer the cosmos with only what we know today, no soft scifi stuff needed. It will just take a very, very long time. Now of course you can't disassemble stars, or for that matter high mass objects like gas giants. But the earth sized planets or at least large portions of their lithospheres can be consumed.
The actual Project Orion documents about nuclear pulse propulsion if anybody wants them.
The Launch Loop - A low cost earth-to-high-orbit launch system LOFSTROM, K H AIAA, SAE, ASME, and ASEE, Joint Propulsion Conference, 21st, Monterey, CA; United States; 8-10 July 1985. 1985