As a college student with a limited budget myself, who started out with a 550D, kit lens, and some old manual primes, I can certainly relate to your situation. Your plan sounds like a fairly solid one, but I have a few suggestions. I personally owned the 50/1.8 I (bought used for $99, sold for $150 a year later) and certainly preferred it to the junkier Mark II, but to be entirely honest if you sold it and bought the optically superior Youngnuo version, you'd save around $100 and to be honest, I'd probably last longer since the newest 50/1.8 I is around 25 years old. But that's up to you.
What I do strongly recommend though is skipping the 28-135 and getting the 55-250 IS instead; while the build quality might not be as good and you loose a little focusing speed, the IQ should not be and it is smaller, lighter, better IS, and ultimately the focal length range is much more useful and complements your 18-55 IS much better, and you save around another $100 depending on where you buy.
Finally, you might want to consider the 18-55 IS over the STM version. The optics in the former are according to most sources just as good if not slightly better, and STM, while slightly quieter, isn't usually reputed to be that much faster than a typical micromotor, and you loose out on any benefits of it since you are using it with such an old body anyway, and save another $40 or so.
Why am I suggesting you save all this money? If you make at least one or two of these swaps, that will more than take care of the costs of a used 550D (Rebel T2i) body, which will likely be worth far more to you then any minor things you loose out on, and you gain a number of benefits with the above substitutions as well. Your 350D will go for up to $100 or perhaps a little less, and the 550D costs $250 right now at Adorama, leaving you with about $150 or so to make up in price difference. And if you really want to keep the Mark I and STM, if you say your budget is $200-300 and you bought the 55-250 instead, you'd still have $200 left over which would more than make up the difference to pay for the 550D body.
Trust me, it will really not disappoint; you get so, so much in exchange. That camera is close to the best you can get in terms of IQ in a crop-format Canon, and it will last you well into your photographic journey whereas the 28-135 will, as others have said, almost certainly not. Plus, if you save by way of my other suggestions, you can start saving up for that sweet 10-18 STM, and once you have that you'll be pretty much set for any sort of photography outside of highly specialized areas.
Whatever you choose, best of luck!
