Ouch, $100 is nothing for decent audio.
Decent monitors start at about $100 each, and you still want a DAC.
Decent amps start at $50 and you won't have decent speakers for it to power with the remaining $50.
There are some 3" monitors under $100 that aren't terrible (Presonus Eris 3.5, Samson M30, Mackie CR3, M-Audio AV32) but these are really compact mixing speakers and wouldn't be ideal for general use unless you're the sort of person who doesn't care about bass too much.
Personally, I'd avoid the no-name monitors on ebay and amazon, it's usually cheap paper cones and bottom-of-the-barrely amplifier circuits. They'll probably sound better than your existing set but maybe not by a whole lot.
If you can afford to wait for a bit and save some cash, the Klipsch Promedia 2.1 is probably the defacto PC speaker upgrade. It's not the most refined or balanced sound you can get for the money but it ticks your boxes and it's affordable(ish) at $160-180 if you hunt around. I think if your focus is action movies and gaming it's very hard to beat for the price.
If you want more clarity, fidelity, and a balanced frequency response for enjoyment of more music genres, then you should be looking to acquire some 5" bookshelf speakers from the used market and a used amp to power them, all run off an external DAC. It doesn't need to be a particularly fancy DAC - the one in your monitor would be fine to start off with, otherwise a $10 USB DAC off amazon is going to be as good, if not better. The point of a USB DAC is not to provide the highest quality DAC (your sub-$300 speakers will still be the lowest common denominator) but to ensure that the DAC isn't in a metal box that's flooded with EMI. This is why Integrated DACs in your monitor, if your monitor has a headphone-out jack, are absolutely fine at your budget. Hell, I've been unable to hear the difference between an Acer monitor DAC and my 'audiophile' DAC even when using $500 of speakers.
At any rate, the best low-budget bookshelf speakers in my subjective opinion are the Micca MB42 - you can get them for $50-60 and if you pair with a cheap class-T amplifier in the 30-50W region you should still fit everything in your budget, just!
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