The relative distances to the planets is fixed immediately by Copernican model, and this is what makes heliocentrism ten thousand times better than geocentrism, even without any known physical cause for the orbits.
The relative distances are fixed from the radius of the epicycle — the epicycle transfers Earth's orbit onto the planet, and the ratio of the epicycle radius (not the angular extent, which also includes the planet's motion along the deferent) to the deferent size in the Copernican interpretation directly gives the ratio of the Earth's orbit to the planet's orbit. The relative size of Venus and Mercury's orbit, relative to the Earth's distance from the sun, is given by the maximum in angle they get away from the sun.
This is not surprising, because the epicycle radius is giving you the parallax from the point of view of the Earth's orbit of the different planets. Once you know the absolute size of Earth's orbit, you know the distance to everything else, which is why the Earth's orbit is called the "Astronomical Unit".
This means that just Brahe's observations are sufficient to fix the entire solar system size except for the absolute scale of the Astronomical unit. The location of all the planets in 3 dimensions is completely determined from the assumption that the Earth's orbit is shared between all of them. The fact that the epicycles all are given by a one-year orbital period for the Earth is Baysian-wise extremely compelling evidence for heliocentrism without anything further to say.
This is why it is not correct to say that geocentrists were somehow justified, or had any valid points, or were anything other than the dimwitted reactionaries that they were. This includes Ptolmey, who buried the heliocentric work of Appolonius for political reasons, although even the most casual astronomer of the era was aware that heliocentrism was correct.
He ignored the radius of the Earth as negligible. His estimates for the angle were from the shape of the shadow the sun casts on the moon, and the difference between this and a straight line when the moon is halfway between full and new is too small to percieve precisely. He fooled himself into thinking he measured a different angle, so his estimate was really only giving a lower bound on the distance to the sun. As a lower bound, it was enough to establish that the sun is larger than the Earth, and this was important, in that it lent strong support to heliocentric models. But it was not an accurate method.
Best Answer
Halley's method requires one to measure the timing of the beginning of the transit and the end of the transit; both pieces of data have to be measured at two places of the Earth's globe whose locations must be known.
The picture by Vermeer, Duckysmokton, Ilia shows that the two places on Earth have differing locations in two different directions (the differences in the distance from the Sun and Venus are too small to be measurable): one of them is parallel to the direction of the transit of Venus and will be reflected in the overall shift of the timing; the other component is transverse to it and it will actually shift the line along which Venus moves and crosses the Sun in the up/down direction i.e. it will make the duration of the transit longer.
Each of these pieces of data – overall shift in the timing arising from one coordinate's difference between the two terrestrial locations – and the difference between the length of the transit – due to the other coordinate – are in principle enough to determine the solar parallax. Because synchronization of clocks at very different locations was difficult centuries ago, I suppose that the latter – the difference between $\Delta t_1$ and $\Delta t_2$ – was probably more useful historically. But we're talking about O(10) minutes differences in both quantities.
The calculation of the parallax from $\Delta t_1$ and $\Delta t_2$ and their difference is a simple exercise in geometry but I want to avoid trigonometric functions here.
At any rate, Halley didn't live to see a proper measurement (the transit occurs about twice a century and the two events are clumped together with a 8-year break in between). The best he could get was 45 angular seconds for the parallax; the right answer is about 8.8 seconds. He knew that his result was very inaccurate. Note that the solar parallax is the angle at which the Earth's radius is seen from the Sun, i.e. the difference in the rays needed to observe the Sun from the Earth's center and/or a point on the Earth disk's surface.
When you convert 8.8 angular seconds to radians, i.e. multiply by $1/3,600\times \pi/180$, you get $4.3\times 10^{-5}$. Now, divide 6378 km by this small number to get about 150 million km for the AU.
Some orders-of-magnitude estimates for the numbers. Venus orbits at 0.7 AU so it's actually closer to the Earth than it is to the Sun during the transit. It means that a shift by 6,000 km up/down on the Earth's side corresponds to about 12,000 km up/down on the Sun's side. So the two horizontal lines crossing the Sun on the picture (places on the solar surface where Venus gets "projected") may be separated by about 12,000 km. Compare it with the solar radius near 700,000 km: you may see that we're shifting the horizontal lines by about 1% of the Sun's radius and the relative difference between $\Delta t_1$ and $\Delta t_2$ will be comparable to 1%, too. The last transit in 2004 took about 6 hours so the difference in the duration at various places is of order 10 minutes.
The 2012 transit of Venus on Tuesday night UTC will take over 6 hours, too; the timing and duration differs by about 7 minutes depending on the location, too.
If you've been dreaming about observing the transit of Venus, don't forget about Tuesday 22:49 night UTC; the following transits will occur in 2117 and 2125. There is a blog version of this answer, too.