TL:DR version: cheap guitar needs work. Fix it or replace it?
Hi - After many years of wishing I could just magically be able to do it, I've finally decided it's time to learn to play guitar. Around 18 months ago someone bought me an electric strat-style guitar so this week I hit youtube fairly intensively and got the truss rod adjusted, the action set, the pickup heights set, etc etc. After all that (and bear in mind I can play at most 5 chords, 6 if you gimme a minute to get the F) it sounds.... ok'ish.
It's really rather noisy (not the good noise, the bad noise) with the volume / gain turned up, and awfully 'twangy' sounding. Furthermore I've found that I have a couple of high spots on some higher frets (same note across multiple frets), and one of the bridge pickup height adjustment screws doesn't - half the pickup appears to be floating. To be fair a bit of research indicates that it's a £70 guitar - I'm not fussed about price but thats just to indicate the components probably arn't top notch. Don't misunderstand - it actually doesn't sound half bad if I get the knobs in the right place but I'd like to know that as I learn I can make it sound good, rather than be hampered by equipment from the get go. I'm sticking it through a Vox Amplug2 AC30 or my PC running Guitar Rig 5 (it came with an amp - that just appears to be a box that only makes a 'brrrrrpppppppt' noise).
So, the question. I need to spend some money but I'm not willing to splurge. Should I level the frets, put some decent (budget) pickups in (Ironstone? Wilkinson?), add some shielding, maybe change the pots and caps, or should I just go get a new guitar? The latter option would ideally be something like the Epiphone Les Paul Special II - only £140 but with excellent reviews and apparently a great starter. If I can make my cheapy-strat into something I'll enjoy playing I'll do the work (assuming I don't arse it up) but I don't want to buy pickups, a crowning file, pots etc only to find it still sounds like crap no matter how much I improve. However, I don't want to blow more money than necessary on something that I may well give up on in 2 months.
Sorry this is a bit long but I really would value the opinion of folks who actually know what they are talking about!
Edit: I'm not trying to get to professional level equipment - I just want a beginner plank that doesn't handicap be by being partially broken. I'm really looking to know if the improvements I've listed will a). do the job, and b). are doable for someone with no real experience.