How can I conclude that I live in a solar system? Well, this is an awkward question and I don't know if it is mathematical enough for MO (I'm sorry if not) but I'll try it: What observations in the coordinate system centered in my fixed position on earth are necessary to conclude that the earth (and the planets) move (approximately) in ellipses around the sun and that earth is rotating around itself? 
 A: Evidence that the Earth is spinning about its axis: the Coriolis effect. The coordinate system fixed to a specific spot on the Earth's surface is not inertial, that is, Newton's first law does not hold.
Evidence that the Earth and other planets are moving about the Sun in elliptical orbits: agreement of the projected elliptical motions with observations in the night sky.
There is no fundamental reason to choose the Sun as the center of the solar system coordinates. Any point will do, including the Earth or any other planet. However, as noted by such luminaries as Copernicus, Galileo, Kepler and other MO respondents, choosing the Sun as the center simplifies things quite a bit.
A: I find the parallax effect 
 parallax
effect especially convincing evidence.  Parallax  is the shifting of lines of sight
due to translation, eg by waiting half an earth year at which point
theory tells we have moved about 16 light minutes around the sun from where we were.
Regarding retrograde motions: as Ilya  said,
  Kepler's 2nd law  closer
planets move faster.  Now draw two circles 
centered at 'Sun' with a point 'Earth'  moving on the inner  circle
and another point 'Mars' moving more slowly but in the same sense, say counterclockwise
on the outer circle.  Drow a line between the two moving points. That line 
indicates how Mars looks, viewed from earth, relative to the distant stars.
How does the line move?  Put the Sun at the origin.  If the order is  Sun-Earth-Mars, 
with Earth and Mars on the   positive
x axis, then the slope of the line is decreasing. But put the order Earth-Sun-Mars
with Earth on the negative x axis, Mars on the positive x -axis.  The slope of said line is now increasing. One is `prograde' the other 'retrograde'.   
Finally, the  explanation of  elliptical  versus circular motion 
is more of an Occam's razor business.  Originally we had
Ptolemy's ''epicycles''-- in essence Fourier series.   Ptolemy had earth at the 
solar system center and  each planet moving on a system of nested  circles,
as in   $z(t) = r_1 e ^{i \omega_1 t} + r_2  e^{i \omega_2 t} + r_3 e^{i \omega_3 t} + ... $, $r_1 > r_2 > \ldots $.  Ptolemy  needed   20 to 30 circles to account for observations.
Kepler  realized that but putting the sun at the 'center' and having  the planets move in slightly eccentric ellipses with focus,
 a bit off from the sun,  sweeping out ''equal areas in equal times'', he could account for all of Ptolemy's data plus Brahe's much more detailed data.  
A: An interesting (and very old) argument in favour of heliocentrism is based on estimates of the relative sizes of the Earth and Sun.  
Actually,  Aristarchus of Samos estimated that the Sun is six to seven times wider than the Earth (and therefore over 200 times more voluminous). These calculations arguably led him to conclude that it made more sense for the Earth to be moving than for the huge Sun to be moving around it.
A: You can use a Foucault pendulum to determine that you are on a rotating planet. If you set it up on the North or South pole it will complete one rotation in one day.
A: Take a look at Terence Tao's pdf slides (4.3 Mb), http://terrytao.files.wordpress.com/2009/09/cosmic-distance-ladder2.pdf. Kepler makes an appearance and there's much more besides.
A: I think that you can write down the motion of planets in any coordinate system you like. Since prehistoric times, astronomers have recorded the motion of planets, and before Kepler everybody used the coordinate system where the earth is fixed. I don't know if they ever wrote down equations of the planets' motion, but they could certainly predict it.
Of course, the equations of motion in this coordinate system would be very messy and inconvenient. The reason that we say that "planets move around the sun" is, IMHO, the fact that in the coordinate system centered at the sun, the equations become so simple and easy to understand. 
The one "specific observation" that, to my mind, shows how confusing and inconvenient to calculate the motion of planets is from an Earth-centered system, is the retrograde motion of planets. When viewed from the earth, planets usually move in one direction in the sky (when viewed at the same time but on different days). Sometimes, however, planets like Mars move backwards in the sky. I think this has to do with the angular velocity of the Earth moving around the sun being greater than that of Mars (anybody has a better explanation?).
A: Stellar aberration, the change in the apparent positions of more or less most of the night sky with the seasons, directly due to the velocity of the earth in its orbit.  Note - aberration, which is related to the idea that raindrops appear to fall slanted in a moving vehicle, is not the same as parallax, e.g. has no relation to the distance of the stellar source, etc. :-)
See http://en.wikipedia.org/wiki/Aberration_of_light .
Predicted by Bradley in 1725.   First measured > 0 by Bessel in 1838, with unpublished successful observations by Henderson 5 years earlier.
See http://thonyc.wordpress.com/2010/01/09/we-know-it-moves-but-can-you-prove-it/  .  
A: Isn't the whole idea why we accept the 'fact' that the Earth revolves around the sum based on how science works? 
You have a theory about how something works, do some experiments, and if you don't get a contradiction, you do not reject the theory. I think that if you investigated any of the theories that placed the Earth at the center of the Universe, you would eventually find a contradiction. When you place the Sun at the center of the Solar System, no contradictions develop ( except for the thing about the behavior of Mercury - ask Eddington and Einstein). 
Actually, when you listen to phyicists about why they accept quantum mechanics, they will say that it is a theory that has never encountered a contradiction over the past eighty years. However, they are still looking to a theory to unit it with general relativity, but that is another story. 
