A few answers:
I'd definitely suggest that yes, go ahead and learn a few solos you like. It'll give you many satisfactions and you'll learn a lot of useful things.
If your strings break that often, it may be that you play a little too hard, or there may be some sharp edges on the bridge saddle, the nut, or even on some frets. This is common on cheap instruments. Next time you chance strings -- or even now -- loosen the strings and see if you can ensure that all the places that come in contact with the strings are smooth and free from sharp edges.
If you guitar has a straight neck and good action (i.e. strings not too far from the fretboard) then you don't need to be in a hurry to change it. Instead, watch some videos that show you how to "adjust the truss rod" (google that) and also how to fine tune the bridge saddles (for best tuning and good action). It's not too hard to do, every guitarist should be able to do at least a decent job with that, and you'll also learn useful things about how you playing is affected by different string actions (i.e. by the change in distance between strings and fretboard).
Lastly, about amps/effects: to a beginner with a limited budget I would suggest a multi-effect processor / amp simulator. There are some cheap ones that are quite decent and versatile. This will allow you to experiment with a lot of different effects -- not just distortion, but chorus, echo, reverb, flanger, compression, etc. and the the sound of a few different types of amps.
You can send the output of the processor to your current amp (clean channel, let the processor do the rest) or even in the headphones or to a laptop. Look around for "guitar multi-effect processor" and see what your budget allows. With a bit of luck you may even find a cheap used one in good conditions -- it's common for people to buy gear and then not use it for one reason or another, so their loss can be your gain...