Hi, cute work which has more potential than what you've fullfilled. You've clearly attempted to compose this in a classicicistic sonata form, the effectiveness of which derives from the appropriateness of the proportions and I think this is something you really need to have a look at in this work.
Your exposition is too small. You state the themes and don't really do much more (development please?), and you also modulate (smooth modulation, I'll give you that) very quickly and dispense with any sort of harmonic tension/conflict. The exposition does fulfil a very particular role in a sonata form; that is, the articulated move from the tonic to another key (typically the dominant in a major piece), which you have observed to a degree. However, this isn't merely a "chore" or "tickbox", it's a harmonic event, your chance as an artist to arouse interest/tension due to the harmonic polarity which you have enveloped, which sorry but you did not achieve; the modulation needs to be more persuasive (Beethoven is worth looking at in regards to this; his transitions were often the most intense point of the exposition).
I was more impressed by your development section, you had some interesting stuff going on here and it looks to me from this that you have potential as a composer. However, considering the size of the exposition (and the recapitulation), this was far too large (it dominates the work far too much); even if we talk solely in terms of harmonic "requirements", you'd need a considerably larger recapitulation/coda to properly resolve the harmonic tensions of the development section, which were also hindered by the weaknesses of your exposition section. The recapitulation (like the exposition) was far too brief, and I'd want much more.
Your writing for string quartet was fine, you played it safe but that's okay at this stage. Your ideas are quite good and there's alot of potential, I think you just need to work on developing ideas and on your handling of structural factors/events. It might be worth you getting some string quartets by Mozart/Haydn/Beethoven (since you seem to be interested in classical stylings) and observing the procedures they use in their sonata form movements.
Not a bad effort, but I think you could do much better; in fact I don't think this is even 30% of what you could achieve if you put your mind to it. I'll be interested to hear your future efforts.
P.S. It might benefit you to read Charles Rosen's "Sonata Forms", it's very interesting for those who're interested in the classical era and may help you as a composer.