When Jupiter formed I assume like the other planets it started as tiny clumps of matter that eventually came together, became gravitationally bound and then eventually captured a lot of gas. I've also heard it was capable of collecting a lot of solid ice due to its distance from the Sun. Anyway, if Jupiter were larger we might be living in a binary star system. So, my question then becomes, did the Sun have a similar beginning to Jupiter and in what way was it different? Did the Sun form around a solid core?
Star formation isn't completely answered, but it is well believed that a solid core is not necessary. However if the sun did form around a planetary-sized solid core we would not know the difference. Due to the very high temperature of the sun, the result is not meaningfully different from colliding with planetary bodies early on (which is plausible given the number of planetary body collisions that are invoked to explain the solar system).
No, the Sun is not thought to have formed around a solid core, and solids would not exist at the temperatures and pressures at the centre of the protosun. The Sun formed simply from the gravitational collapse of a large cloud of gas.
The situation for Jupiter is different because far out in the circumstellar disc of the forming solar system, it was cool enough for the condensation of solids. The core accretion model is where gas giant planets begin their lives through the growth of a small (well maybe 10 Earth masses!) rocky/icy core, this is later followed by a brief and rapid accretion of the gaseous envelope.
It is currently not known whether Jupiter has a rocky core or not. That is one of the key questions that it is hoped the Juno mission will answer.
Recent years have seen the re-emergence of the thermal instability model for the rapid formation of gas giants. Such giants would not have a solid core and this formation mechanism is more akin to the way that the Sun formed.
As to whether there could be observational evidence that the Sun did not form around a solid core, I am doubtful. A solid core would of course vapourise at the high temperatures inside the Sun, but its enhanced mean atomic mass could remain, altering the structure of the core, nuclear reaction rates etc. The trouble is, that the vaporisation would occur whilst the contracting Sun was still fully convective, effectively mixing everything up throughout the Sun before nuclear reactions began.
The Sun did not form around a solid core. Rather, it seems to have formed from a cloud of collapsing gas that may have been further enriched by matter from a nearby supernova. Gravitational force caused the collapsing cloud to start spinning, and the spinning compressed it into a disc with a bulge in the center that became the Sun. Here is a better explanation: http://image.gsfc.nasa.gov/poetry/ask/a11379.html.
The sun definitely does not have a solid core. The temperature and pressure is way too high to maintain such a close-proximity atomic structure, especially since it is a hydrogen fusion reactor. Solids have atoms with nuclei that remain in a constant position (not like, the object doesn't move, but it's relative position is fixed), but that would mean that fusion could not take place (because the nuclei wouldn't fuse).
Matter (gases and dust) were pressed together by waves of energy in space and gravitational force pulled more and more matter together (it doesn't really matter where the matter came from, old star remnants, supernovas, miscellaneous mass, etc.). Gravity caused spin on the clouds of matter and dust, which made it sort of flat like a disk and it clumped together (from the high temperature and pressure) into a protostar.
As explained above, the dominant theory of stellar formation, and thus the formation of the Sun, is through the collapse of gas. The theory goes that a large cloud called the giant molecular cloud will start "clumping" through a combination of gravity and shock waves, and these clumps will eventually collapse into stars. A giant molecular cloud can form many stars simultaneously.
The answer to your question of why the Sun didn't form from a planetary "solid core" (they are called planetesimals) is related to lengthscales. There's a hierarchy in astrophysical formation processes where larger bodies precede smaller ones. The hierarchy starts first with the giant molecular cloud which is an amorphous blob of gas. This large cloud will break into smaller clumps that will eventually collapse into collapse into stars. And then, the leftover matter of these clumps around the stars will start forming planetesimals that will lead to planets (the planetesimal forming matter around this stars is called a proto-planetary disk). A way to think of it is how you can't really have the branches first before the tree. So the order is molecular cloud -> stars -> planetesimals -> planets.