Ground → ship Wi-Fi bandwidth in my fast moving spaceship I have a hypothetical question that hopefully makes sense considering only the rudimentary amount of knowledge I have on relativity.
Assume I am in a spaceship orbiting around a spherical satellite, at a certain fixed radius.  The satellite sends data to the ship via radio (something like satellite internet or other long-range data radio, using electromagnetic signals).
Would the download speed (received data rate seen on-board ship) be faster or slower, compared to when I am stationary with respect to the satellite?
Also consider that attenuation is negligible.
What I initially thought was that time dilation would make "my" clock in the spaceship tick slower, hence the signal from the "satellite" would be received faster. I thought I should add this to clarify my train of thought that led to the question. Please do note that it is a hypothetical question.
(A comment pointed out that a spherically symmetric EM wave is impossible, therefore I have taken down the diagram and the statement regarding so.)
 A: Doppler effect (relativistic) of the Electromagnetic Waves will take place. As a result you will receive the emitted Wifi signal (EM wave) as squeezed along the time axis (increases frequency). Naturally the WiFi demodulator available with you would report invalid (unrecognised) signal because digital data is encoded on the carrier wave by some form of modulation like Frequency Shift Keying , Phase shift keying etc. 
Let's say it's transmitted by FSK. In FSK, the 0s and 1s are transmitted through distinct changes in Carrier Wave frequency held for a particular duration. First of all the EM wave frequency has changed  hence your demodulator won't even recognise the EM wave as Wifi signal. 
Assume that you have accounted for Doppler effect EM wave frequency change and distinct change in the changes of the changed frequency (obtained from it's corresponding changed Fourier transform) for zero and one encoding, into your custom demodulator. So the EM wave is now recognised as Wifi signal. But as stated above the distinct changes in EM wave frequency is held for a pre agreed time (between transmitter and receiver) to encode 0 & 1. So due to time dilation this agreement breaks. So your demodulator although recognises Wifi signal presence, reports invalid protocol. 
Now you might be tempted to think let's account for the time dilation into the demodulator. But that isn't really a good idea because the above stated pre agreed duration was statistically obtained to ensure noise free transmission. So if you try to account for that in your demodulator, you will occasionally receive corrupted data due to noise in the transmission. But if you are okay with it, then sure download speed increases!!!! 
A: If the satellite's signal is broadcast from the centre of the satellite, so your fixed-radius orbit doesn't alter the distance between you and the antenna, you'll get relativistic doppler shift:
According to "The relativistic Doppler shift in satellite tracking"

Navigation Technology Satellite 2 carries aboard it two
  high‐precision cesium beam atomic standards.[...] This observed bias
  was 4.47 parts in 10^10, which is consistent with a predicted
  frequency bias arising from the special relativity time dilation
  effect coupled with a frequency blue shift predicted by general
  relativity.

I would describe that as a relativity-induced error of 0.447 parts per billion
On the other hand, if the antenna was on the satellite's surface, you'd also see doppler shift as the distance between you and the antenna varies.
According to "High-Accuracy Prediction and Measurement of Lunar Echoes" 

The relevant rates of change are usually dominated by Earth rotation, which at the equator amounts to about 460 m/s. As a consequence, two-way Doppler shifts can be as large as ± 440 Hz at 144 MHz, ± 4 kHz at 1296 MHz, and ± 30 kHz at 10 GHz.

So, if your satellite is the size of earth, you might see a doppler shift varying between +3000 parts per billion and -3000 parts per billion over the course of an orbit.
However, communication standards already expect a certain amount of doppler shift - according to "Next Generation Wireless LANs: 802.11n and 802.11ac"

in 802.11n [...] the values for Doppler spread in the 5GHz band is approximately 6Hz

So the makers of the standard already expect a doppler shift of 1.2 parts per billion. Cell phone mobile data standards like 5G worry about doppler even more - allowing them to work even on high-speed trains.
But meanwhile, according to "Clock Solutions for WiFi (IEEE 802.11)" (admittedly a dated source)

The typical performance requirement is ±25ppM all-inclusive frequency
  stability (includes initial calibration tolerance at 25°C and
  frequency changes over operating temperature, power and load
  fluctuations, and aging).

So WiFi hardware makers are already compensating for a 25,000 parts per billion crystal frequency error, which can vary with changing temperature conditions.
I would say, as long as any motion-induced doppler frequency changes are small and slow compared to the thermally-induced crystal frequency changes wifi can already account for, you should get normal performance from a frequency perspective.
Of course, depending on your orbital radius you might encounter problems due to signal latency - and inverse-square-law signal attenuation. And the signal attenuation happens in both directions, so even if the satellite has a megawatt transmitter to give a really long range, you won't get a two-way link unless your wifi hotspot has the same great range.
A: If you are in a perfectly circular orbit, then your received signals are affected by the relativistic transverse Doppler effect.
The signals you would receive would be blueshifted by the Lorentz factor of the orbital speed. That is the information transferal rate would (potentially) go up (if you can tune to the new frequency).
If you are in a Keplerian orbit, then (in a Newtonian approximation)
$$ \frac{mv^2}{r} = G\frac{Mm}{r^2}$$
$$\frac{v}{c} =\sqrt{\frac{GM}{rc^2}} = \sqrt{\frac{r_s}{2r}},$$
where $r_s = 2GM/c^2$.
The transverse Dopper blueshift is by a factor of
$$ f_r = f_t\gamma = f_t\left(1 - \frac{v^2}{c^2}\right)^{-1/2} = f_t\left(1 - \frac{r_s}{2r}\right)^{-1/2},$$
where $f_r$ and $f_t$ are the received and transmitted frequencies - assuming that the transmitter is at $r=0$ or at least stationary with respect to the orbital speed.
However, in General Relativity there is also a gravitational redshift given by 
$$ f_r = f_t \left( \frac{1 - r_s/r}{1 - r_s/r_t}\right)^{1/2},$$
where $r_t$ is the position of the transmitter (which has to be at $r_t > r_s$) and this may modify the conclusion.
Which will win out depends on the radius of the orbit compared with the radius of any "surface" you put your transmitter on. 
A: There are two effects due to relativity.
The motion of the spaceship around the satellite results in a time dilation that can be calculated using special relativity.  
If the satellite has significant mass (e.g., enough to keep the spaceship in orbit), then general relativity (gravity) plays a role.
From the spaceship's perspective, the two effects are opposite, but not equal in magnitude.
This Wikipedia article re error correction of GPS signals may help.
Per the suggestion by @ChappoSaysReinstateMonica, the following is included here as an easy way to think about it:

Assuming the satellite has essentially no mass, so the spaceship needs to use its engines to keep going in a circular orbit around the satellite, then only special relativity applies. The spacecraft will perceive its received data rate as being a bit faster than the transmitter's data rate on the satellite. A way to know this is via the fact that a clock taken from the satellite to the spacecraft, then after a while brought back to the satellite, will lose a bit of time relative to a clock left on the satellite, but uploaded data is like clock ticks on the satellite. 

A: Basically your question is the same as what Michaelson and Morley asked. Their experiment used the earth traveling around the sun instead of a spaceship. Their results did not show a difference in the speed of light either way.
