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
radio waves emitted by a radar the probe carries, reflected by the surface of some target
the communication signal between probe and Earth (see for instance, the deep space network).
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.
Galileo Galilei had a similar question like yours. Only, back then he was asking how to measure the speed of the ship on which you are moving.
He came to the following conclusion - you cannot measure the speed of the ship if you don't look outside the ship - to the stars, passing islands and so on. This is what we now call Galileo's principle of relativity.
Back to your qestion - if there is nothing around you and you also don't see any stars from your spaceship (because of dust around) and your gadgets don't 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 it's always the ground.
If I have a ship and I'm 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.
One way to measure the speed of an object in space is to see how fast it moves against a lit background. For example, Mercury moving in front of the sun. We could see a black blob (Mercury) move across the bright sun, and record how long it took. Then we could say that it takes $x$ amount of seconds for Mercury to cross over the sun. If we also knew more information on their sizes then we could get that speed of Mercury into km/hr. This site has more analytical ways to calculate that if you're interested in how they get certain numbers.
Furthermore, if we are on a rocket ship and flying through space, we could look out the window and do the same thing in a sense by looking at planets and suns and seeing how fast we pass them. Or, we could still get km/hr even if the rocket ship had no windows. If we know how much fuel we're exhausting over time, we could calculate that as km/hr. It's like the car, it has a built in system to determine how fast you are going ... it doesn't shoot a laser at the passing scenery to determine speed.
One possibility is to measure your speed relative to the ions around you. Aeroplanes in the early days had the same problem. They used ventura pipes where the drop in air pressure was a function of speed. This was air speed not ground speed. A tail wind meant their air speed was in accurate. They did spot checks on land marks to Guage the air speed. Space is full of ions. Don't know how but maybe if these ions passed through a copper coil it would induce a current. Another possibility is to put multiple GPS salellites around the sun at different distances and sending the signal in 360 degrees.