Not really a problem to make a nuclear powered star ship, in 1960's it was already solid tech and expected speed of around 1/10 C is quite possible..
The problem is the amount of fuel needed to power it, an accident on route to fuel it could be a global disaster.
Creating the ship and fuel at Mars is a different story, but we are some long number of years before we are ready to do that.
With a 45-50 year trip to the nearest star, a refuel of a few years and creation of a second ship and exploration of planets 20-30 years, then another 45-50 year trip to the next star we could explore the entire galaxy in 1-2 million years.
Robots best suited to these trips and if we duplicate more ships at each destination far less than 1-2 million years.
At 50-55 years we get data back about the first star and planets.
Next 70 years we get data back about 2 stars and planets, 70 more years and it's 4, etc etc.
Smallest possible ships, smallest possible amount of fuel and easiest duplication i think is the ticket to explore the galaxy.
Getting to the next galaxies a far longer prospect but possible also.