Would a spacecraft just go "through" a gas giant? From my understanding of the word gas giant, it is a planet composed of entirely a gaseous atmosphere, and so planets Jupiter and onwards fall in this category. 

That being said, what would stop a spacecraft from just going right through any of these planets?

For earth, a spacecraft would firstly rapidly disintegrate approaching the surface of earth, and if it happens to still have part of itself, it will collide with the surface.

For gas giants like Uranus, if a spacecraft is able to counter the gravitational forces and slow down, could it actually "go through" Uranus?
In short, No.

Side detail:
Uranus and Neptune consist likely of 20% gas and 80% rock, coming from simple density considerations. They have large inner cores with masses around $\rm 12-14 \; m_{earth} $, and something like $\rm 2-3 \; m_{earth} $ of gas on top of them.
Jupiter and Saturn are true gas giants. They both have $\rm 5-20 \; m_{earth} $ of solids in them  and the rest of the $\rm 320 \; m_{earth} $ (=Jupiter) and $\rm 95 \; m_{earth} $ (=Saturn) are gas.  

However even if Jupiter and Saturn were 100% gas, you couldn't fly through them.

The main reasons for this is


Enormous density, pressure and temperature. A gas mass of several tens of Earth masses will collapse under its own gravity to enormous central densities, temperatures and thus static pressures.
The densities that this gas will be compressed to, can have densities comparable and exceeding that of water, which is a good rule-of-thumb for the center of Jupiter (typical calculations show gas densities around $\rm 20 g/cm^3$, source).
The static pressure will crush your spacecraft, the high density will near instantly slow you down to zero due to strong friction, and the high temperatures will just melt or evaporate your spacecraft.  


Just because something is gaseous, doesn't mean you can soar through it, as you would on a calm day with a glider on a light breeze.

There are other factors that complicate even getting there:  


Re-entry heat. The cold upper atmospheres of the gas giants increase the mach numbers $v / c_{sound}$ at re-entry by up to a factor of two. As recently discussed on space exploration the re-entry heating that a spacecraft experiences scales as the 8th power of the re-entry speed or more. So a spacecraft would experience around $2^8 = 256$ times more heat load. The engineering challenge to survive this already would be considerable.
I bring this up, because as @uhoh pointed out, it is unclear whether you want to just slice the atmosphere or fly through the $r=0$ coordinate of the planet.    
Core erosion. Funny enough, the solid core is probably not even there anymore. Recent work has shown, that under the enormous pressures given at the center of a gas giant a phenomenon called 'core erosion' will take place. In this, typical rock-composing elements prefer to exist in a gaseous phase rather than in a solid phase at those high pressure, thus the solid core will dissolve into the surrounding gas.
This will further increase friction when trying to fly through the gas.