How is the speed of an object in space measured? Also more importantly how do you measure your own speed in space? On the road we use a speedometer which tells us the speed easily. How is it done in space?
The main question is "relative to what?"
For space probes and the like, the speeds that matter are be either with respect to the Earth, the target object(s) (Mars, some asteroid, Space station, etc.), and/or the Sun (or Solar system barycenter). These speeds are measured mostly by Doppler shifts in
Other methods have been used (image analysis between consecutive images taken by the space probe, the temperature of the heat shield on atmospheric entry, etc.) but these are all much less precise than Doppler measurements.
Space telescopes will measure redshift to some object (star, galaxy, etc.) (which is very similar to Doppler), which is more an indication of how fast that object is moving with respect to the entire solar system, rather than just the space telescope. Parallax methods are also used (see @IgnacioVazquez-Abrams's answer), but such methods can only be used for objects relatively close by (the parallax for most galaxies is too small to measure).
Other methods include Cepheid variables, and of course the famous Type 1a supernovae, which were used to conclude that the expansion of the universe is accelerating. But these are primarily measures of distance, and only crude measures of speed -- for objects at large distances, redshift is the only accurate way to measure the speed with respect to those objects.
Parallax with various objects is measured, and known distances are used to triangulate your position in 3 dimensions at two moments in time. Since you have both the space displacement and the time displacement, you can just divide to calculate the speed.
Galileo Galilei had similar question like you. Only back then he was asking how to measure the speed of the ship on which you are moving.
He came to the following answer - you CAN NOT measure the speed of the ship if you dont look outside the ship - to the stars, passing islands and so on. This is what we now call the Galileo's principle of relativity.
Back to your qestion - if there is nothing around you and you also dont see any stars from your space ship (because of dust around) and your gadgets dont register any fields around the ship (magnetic and electric) there is NO way to measure your speed :)
Relativity means that SPEED as a concept always needs somebody else , somebody with relation to whom this SPEED is calculated - in practical life its always ground.
If i have a ship and im moving there is no speed yet, speed appears when my ship is moving through something - planetary system, system of stars, galaxy and so on.
To sum up the SPEED as well as many other things in life is always RELATIVE and NOT ABSOLUTE.
Thank you for thinking about this message.