So far all answers have got the effects of relativity wrong resulting in incorrect and backwards results.
The principle of relativity states that there is no privileged reference frame, and that the laws of physics take the same form in all inertial reference frames (i.e. the reference frames of free-falling objects).
It follows from this that for a free-falling spaceship (i.e. one not being significantly slowed by interstellar drag, and not accelerating with its engines etc) passing by a planet at a significant fraction of the speed of light the spaceship will see the planet's clocks speed up or slow down by the exact same factor that the people on the planet will see the spaceship's clocks speed up or slow down.
One of the other postulates of relativity is that light always travels at the same speed in all reference frames.
From these two facts together we get length contraction and time dilation.
Length contraction means that the people on the planet see the ship compressed along its length (and by the principle of relativity, people on the ship see the planet compressed along the direction of travel).
Time dilation means that the people on the planet see the ship's clocks tick slower than their own (and by the principle of relativity, people on the ship see the planet's clocks ticking slower than their own).
The relevant factor here is the Lorentz factor: γ=1/sqrt(1-v^2/c^2)
This factor is always greater than or equal to 1.
An observer on the spaceship will count this many ticks of their own clock for every tick of the planet's clock, whilst by the principle of relativity an observer on the planet will count this many ticks of their own clock for every tick of the spaceship's clock.
Conversely, the observer on the spaceship will see this many of the planet's metre rulers fitting within the same distance as one of their own (on the spaceship), and of course, vice versa for the observer on the planet.
This appears paradoxical because we are used to living in a Newtonian world where time and distance are absolute quantities, but this is not the case in Relativity.
Most of these paradoxes can be resolved by giving up such Newtonian ideas.
Now for other effects.
Relativistic mass isn't really a concept used much these days, as it leads to incorrect assumptions down the road. Instead, we include the Lorentz factor explicitly, with E=mγc^2, p=mγv etc. Unfortunately pop-science books and TV shows love it.
The main other noticeable effects would be the Relativistic Doppler Effect & Relativistic Aberration. The Doppler effect is similar in relativity to that of Newtonian mechanics, so you'd see objects ahead of you look bluer (or depending on the speed, possibly shifted into the ultraviolet or gamma spectrum) than normal, and ones behind look redder (again depending on the speed, possibly being shifted outside the visible range into the infrared or radio spectrum), but unlike the classical effect, objects perceived as directly to your side will also appear redshifted. The diagram below shows how the colour of uniform distant yellow stars will appear in different directions for an observer moving to the right with the Relativistic Doppler Effect applied on the top, and only the classical one on the bottom.
Additionally, Relativistic Aberration (which is closely related to the Relativistic Doppler Effect) will cause objects to appear to be closer to the point directly in line with the direction of travel (i.e. directly in front, or directly behind).
By the principle of relativity, the observers on the planet will also see the spaceship to be relativistically Doppler shifted (including being redshifted as it moves perpendicular to the planet), and to appear closer to the point directly in line with the direction of travel (i.e. further away than it is as it approaches, and nearer than it is as it departs).
Note: so far I have only considered Special Relativity. This is sufficient to cover the case described in the OP, as special-relativistic effects will be much stronger than any general-relativistic ones. Special relativity applies to any motion on a "flat" spacetime (one where the gravity is weak compared to the scale of the experiment). If you start going near black holes or doing long-term observations at low speeds in gravitational wells you'll start getting other effects coming in (e.g. gravitational time dilation where clocks lower down a gravity well tick more slowly than those higher up).