Improvement to car sound will always be welcome but I'm finding, even in GT5, that a good headphones and headphone amps make a world of difference. For around $60 (even less from Amazon), you can get a decent amp like the FIFO E11 that can drive pretty much any headphones to a decent level and add a nice hump to the lower midrange and upper bass. I'm using mine with the Sony MDR-7506 (the standard studio headphone).
Don't let the photo fool you, it's a small amp (much smaller than a portable HDD):
I wouldn't say that's good advice.
Since the following is a bit long and maybe complex,
tl;dr: Save that money and buy the AKGs recommended by helseth.
Prevention: I'll stick to the hardware side of things, as I cannot commentate much about game sounds and samples in particular (software, recording, mixing), so stick with reading Griffith in that part.
First off, any gaming or music enthusiast agrees on that what matters the most, by far, are a good set of speakers or headphones. DACs are secondary at best, so in general terms money should not go into them. Also, Amps aren't needed or a nice benefit for most headphones.
Second of all, in general terms music focused products aren't as good as some of the legends for gaming, no matter price range, or even some gaming focused products. In headphones and in speakers there's some tradeoffs to be made, hence why no headphone and no speaker can be perfect in all regards, some of which are quite a nice addition for gaming, casual or competitive.
Thirdly, while great sound systems make games alive, even the old ones and those that are known for having not great samples, still GT6 (the current state) will not sound as good as other games through the same set.
Finally, the 'world of difference' comes from using something different than the TV speakers and from playing a different game, so lets say a cheap HT or "old" stereo in a simbin sim, but from there there's only nice improvements even when going to the very top of the chain (the upper AKGs), not more. Maybe the only breaking exception in this is one specific ATH, but only for competitive gaming because they are pretty bad for enjoying the game.
So, with all that said:
1.- Don't run GT6 through the TV speakers. Just don't.
2.- Now on the higher end and complicated stuff, if you want to,
a) Headphones: Save that DAC money and buy the headphones helseth recommended, the AKGs, as those are world class for gaming. If not interested in that, which is the majority really, then buy a nice home theater. The issue can get a bit more complicated on both based on impedance, sensitivity, passive speakers and whatnot, so the general advice is to buy the simple stuff that works; no amp needed or that provides substantial benefit.
b) As for DACs in particular, now on the bottom of the chain in this part, nowadays in gaming no one uses 2.0 DACs but 7.1, those dolby surround, creative, etc. technologies, particularly for competing in FPS'. Those are pretty good for casual as well. Be sure to change game settings to suit that (7.1, surround, large theater).
3.- And now about FIIO, the very bottom of importance, as far as I know and have tried FIIO products are pretty awful for gaming, to the point for competing it's better to use the integrated sound card. Tried pretty much all its upper range.
And you know what, in headphones I think what's best is to start with a pair that does the job and without buying nothing else, for example some superlux, then do a big climb on the ladder, buying expensive headphones, good dac, amp, etc., and FIIO products aren't really considered at the "audiophile" level (music) so I can't recommend them at all unless if the only advantages they have above others is what matters the most to you: cheap (though will end buying another DAC/Amp soon) and portable (gaming isn't that much).