Is it possible to travel near light speed and not notice relativity? Suppose I have no idea about relativity and find an ancient spaceship that takes me to Proxima Centauri in just 1 year more than it takes light to make the trip (so some observer might say it took about 5 years).
Now from my point of view it only took 1 year (if it had been at light speed, the trip would have been instant). My new friends at Proxima Centauri also agree that they saw it take 1 year (the light from Earth when I left reached them just 1 year before they saw me arrive). None of us could perceive anything other than classical Newtonian mechanics happening.
Meanwhile others on Earth who never understood relativity would observe that my trip took 9 years and the photos I send to them upon my arrival show I only aged 1 year. They would be surprised.
Is this correct? Meaning for a naïve space traveler from Earth and like-minded observers at the destination, relativity wouldn't seem to set any speed limit?
 A: The part where you say the traveler wouldn't see any set speed limit is correct.  They could always go faster and get there in less time in their frame of reference.
An observer who sees the traveler moving would never see them move at or faster than the speed of light.
A: 
Now from my point of view it only took 1 year (if it had been at light speed, the trip would have been instant). My new friends at Proxima Centauri also agree that they saw it take 1 year (the light from Earth when I left reached them just 1 year before they saw me arrive).

No, they'd see you take 0.1 years.
For the ~4.5 year trip to appear to the spaceship inhabitants to take 1 year, time dilation has to be 4.5. That happens at 0.975c. This means your spaceship would arrive about 0.1 years behind the light of you taking off from Earth. The people on Proxima would see your ship take off, seemingly approach faster than the speed of light as the distance between your ship and them decreases, and then land 0.1 years later.
The spaceship inhabitants would think they took one year, the Proxima inhabitants would say they just saw you take off 0.1 years ago, and you'd know something was wrong.  Also the people on Proxima would see some crazy optical effects, like your ship would appear to be elongated and a bit distorted (distorted because it's not traveling directly at Proxima but where it will be in 4.5 years) through a combination of Lorenz transformations and optical effects.
If the people on the ship observe Proxima on their approach they will see similar effects, to the observer on the ship they are stationary and Proxima is approaching them at 0.975c!  Everything on Proxima will appear to be time dilated and it will appear elongated and distorted.  As the ship get near and slows, these effects will diminish.
Here's a great site visualizing the optical effects on observing objects moving near the speed of light.
A: Easy to make simple mistakes in time dilation. M.M. Is right (on the time, the rotation is irrelevant). 
The dilation factor is 4.5, it means both your friends back on Earth and your new friends in Proxima saw you take about 4.6 years, whereas you felt or aged only 1 year. Proxima and earth people are in the same coordinate frame (with their x=0 offset by 4.5 ly, but they can do that classical Galilean transformation). They are at rest wrt each other. They measure time the same way. The people in Proxima would know when they saw the light of you taking off from Earth .1 years before you arrived that the light took 4.5 years, and you took 4.6 years. They say 4.6, you say 1, you know there is a relativity effect. 
