Relative motion at almost the speed of light My title might not be precise but I didn't know how to decribe it better.
I've been reading about the special and general theories of relativity for a short while and I started wondering. First, every movement is only relative to some other object/observer, there is no absolute movement/speed, correct? Assuming so, if an object moves at almost the speed of light relative to me, do I also move at almost the speed of light relative to it? If so, am I affected by all the "weird" effects that follow moving at such speeds (increased mass, different time flow, whatnot...)?
From everyday observation I'd have to say not but in that case what defines "moving at almost the speed of light"? Does it have to be relative to something (and what)? If not, is there any absolute measure of speed?
I know I might be talking nonsense but I'm new to these concepts and I'm trying to understand.
 A: The key point here is the "weird" change in the values you measure depends on what frame you observe them in relative to the object. There is no experiment you can conduct to determine an absolute value for your speed. Speed only makes sense relative to an object. So if a near light speed object relative to us was heading towards us, from its perspective they would measure very different values for the length of objects and time for events to occur etc, and likewise we would measure these values for it as different than in its rest frame. 
A: 
If an object moves at almost the speed of light relative to me, do I also move at almost the speed of light relative to it?

Yes, when the motion  of the two frames moving relative to each other is considered to be in one inertial frame, i.e. no accelerations .

If so, am I affected by all the "weird" effects that follow moving at such speeds (increased mass, different time flow, whatnot...)?

No. All  the weird effects modeled successfully by special relativity , are seen , when observed  from one frame , the observer frame, to exist in the observed frame. So if you alternate the observer frame, each sees "weird" effects in the other frame.
A: Yes, all speed is relative, so you do not have a single speed- instead, you simultaneously have all possible speeds relative to all possible reference frames. You are, right now and always, exhibiting all those 'weird' effects in other reference frames. There are frames in which your heart is now beating once an hour, in which your body is as thin as a sheet of paper but expands to its normal width when you lie down and shrink to centimetre long.
As far as increasing mass is concerned, what the effect means is that it takes an increasing force to accelerate you at a given rate as your speed increases, so if you assumed Newton's second law (f=ma) always applied you would have to conclude that your mass had increased- what it really means is that the second law is only an approximation valid at low speeds. In reference frames in which you are moving close to the speed of light it would take enormous energy to increase your speed by a metre per second, even though in your rest frame you can simply start walking at that speed.
