I know that your 75-300 has IS but as you correctly guessed the 55-250 has third gen. IS, which is supposed to give a 4-stop advantage. On the other hand 75-300 is the first gen.
As I mentioned, the 55-250 is reasonably sharp (of course it would be silly to compare it to the 70-200 L lenses). It was similar to the 70-300 non-L I owned at the time, which is pretty good up to 200mm.
Finally while the autofocus is slower than ring USM, I guarantee you will find huge improvement over the antique DC motor in the 75-300 IS. And the AF might be slower due to the slower motor but I didn't find any it to be inaccurate or any hunting in normal light. Of course, you wouldn't even use it in low light.
So, that's my 2 cents. I wholeheartedly recommend it to my APS-C user friends who are not looking for L-lenses and don't have the "won't buy EF-S because I will upgrade to FF soon" syndrome.
I agree, the EF-S 55-250mm f/4-5.6 IS (I or II) is not a bad lens at all. For a good kit on a tight budget I recommend this lens, and the matching 18-55 IS (I or II) plus possibly the 35 mm f/2 for low-licht stuff. All these lenses are best equipped with a good filter and the original hood to keep stray light and dirty fingers out of the front element and filter
Don't buy the 18-55 IS second hand unless you're sure it's seen little mileage: the ribbon cable for the AF motor will fatigue with every zoom action and break eventually, making it a manual focus lens.
5D3, 5D2, Sony α6000, S90 | SY14 f/2.8, Ʃ20 f/1.8, 24 f/2.8, 35 f/2, Ʃ35 f/1.4A, 50 f/1.8 I, Ʃ50 f/1.4 EX, 100L Macro, 17-40L, 24-105L, 70-200 f/2.8L IS II, 1.4x II, 70-300L, 100-400L | E-mount: SY12 f/2, Ʃ19 & 30 f/2.8 EX DN, 16-70 ZA OSS, 55-210 OSS, Metabones SB | FT-QL, AE-1P | FD(n) & FL lenses