This question cannot really be answered because you cannot travel at the speed of light. See Accelerating particles to speeds infinitesimally close to the speed of light?
If you were massless, you would always travel at the speed of light. However, in that case you would not perceive the passing of time. In relativity, the time that passes for an observer depends on the proper time. The proper time for a light-like trajectory is always zero, so photons themselves do not experience the passage of time.
If you travel very near to the speed of light - perhaps 99.9% light speed relative to Earth, you would still be able to view yourself normally in a mirror you carried with you. That is ensured by the principle of relativity, which states that all physical processes work the same way at any constant speed.
The most obvious experimental signature of tachyons would be motion at speeds greater than $c$. Negative results were reported by Murthy and later in 1988 by Clay, who studied showers of particles created in the earth's atmosphere by cosmic rays, looking for precursor particles that arrived before the first gamma rays. One could also look for particles with spacelike energy-momentum vectors. Alvager and Erman, in a 1965 experiment, studied the radioactive decay of thulium-170, and found that no such particles were emitted at the level of 1 per 10,000 decays.
Some subatomic particles, such as dark matter and neutrinos, don't interact strongly with matter, and are therefore difficult to detect directly. It's possible that tachyons exist but don't interact strongly with matter, in which case they would not have been detectable in the experiments described above. In this scenario, it might still be possible to infer their existence indirectly through missing energy-momentum in nuclear reactions. This is how the neutrino was first discovered. An accelerator experiment by Baltay in 1970 searched for reactions in which the missing energy-momentum was spacelike, and found no such events. They put an upper limit of 1 in 1,000 on the probability of such reactions under their experimental conditions.
For a long time after the discovery of the neutrino, very little was known about its mass, so it was consistent with the experimental evidence to imagine that one or more species of neutrinos were tachyons, and Chodos et al. made such speculations in 1985. In a 2011 experiment at CERN, neutrinos were believed to have been seen moving at a speed slightly greater than $c$. The experiment turned out to be a mistake, but if it had been correct, then it would have proved that neutrinos were tachyons. An experiment called KATRIN, currently nearing the start of operation at Karlsruhe, will provide the first direct measurement of the mass of the neutrino, by measuring very precisely the missing energy-momentum in the decay of hydrogen-3.
References
Alvager and Kreisler, "Quest for Faster-Than-Light Particles," Phys. Rev. 171 (1968) 1357, doi:10.1103/PhysRev.171.1357, https://sci-hub.tw/10.1103/PhysRev.171.1357
Baltay, C., G. Feinberg, N. Yeh, and R. Linsker, 1970: Search for uncharged faster-than-light particles. Phys. Rev. D, 1, 759-770, doi:10.1103/PhysRevD.1.759, https://sci-hub.tw/10.1103/PhysRevD.1.759
Chodos and Kostelecky, "Nuclear Null Tests for Spacelike Neutrinos," https://arxiv.org/abs/hep-ph/9409404
Clay, A search for tachyons in cosmic ray showers, http://adsabs.harvard.edu/full/1988AuJPh..41...93C Australian Journal of Physics (ISSN 0004-9506), vol. 41, no. 1, 1988, p. 93-99.
Best Answer
If a tachyon starts from where you are and goes away at faster than the speed of light, you will see the photons it emits earlier than it actually departs. So you will see all these photons coming as if the tachyon were coming toward you at a speed slower than light, and then bang, the tachyon leaves. In fact, the faster it is going away, the slower it appears to be arriving.
EDIT: You can just tell this from a space-time diagram:
Here, the time T axis is vertical and the space X axis is horizontal. Line C represents the speed of light. Photons move parallel to that line.
If something is moving away from you slower than light, it is a diagonal line falling in the slow (s) region. When it emits photons, they travel parallel to C, so each one arrives back at you at a later time. That's the normal behavior that you're used to.
If something is moving away from you faster than light, it is a diagonal line in the fast (f) region. When it emits photons, they travel parallel to C, and thus arrive back to you at a negative time, relative to when the object left you. In fact the faster it's moving (closer to horizontal) the earlier its photons will arrive (negative T). The slower it's moving (closer to C) the more its photons will appear to come all at once, just before it "departs".