Practice amps are just that. They don't tend to ave much character which is what amps are actually about. This is why I mention acoustic. Acoustics are better for learning fundamentals and developing dexterity that will translate to enhanced electric playing (that is, if you can do it on acoustic you can do it even better on electric). Financially, one can do very well for cheap on acoustics whereas one must be very careful to do so well with cheap electrics.
I have carried my electric and used my iPad as a headphone interface and apps to do effect processing and it works quite well so if that is really your thing then look into that (if you have an iOS device that is). The only reason I would carry my electric though is because I expect significant environmental changes that I'd rather not subject my acoustics to. Generally, speaking, acoustics are far better travel guitars because all you need are it and a pick (and even that is optional).
So, yeah, ultimately I would push someone towards getting an acoustic and an electric, lol.
One option to consider would be getting a multi-effects pedal like say a Zoom pedal that has headphone out on board. As Awaclus alluded to, I've found very little utility with headphone playing.
$300 can land you a solid electric or acoustic if you choose wisely. $500 widens the quality margin a whole lot. Once you break the $700-$800 mark you are generally not having to worry so much about quality and are going for specific features you desire which generally means you know what you want and what rigs are going to offer that.
A decent amp, depending on your desires is going to often run in the $300-$500 range. Now, I bought my favorite amp in 1995 for $60 at a pawn shop. It's an early 70's Fender Vibrochamp. I've since swapped out the tubes and replaced the speaker to the tune of maybe $100 but it's worth in the neighborhood of $700-$800 and sounds amazing. It was a steal even then in 1995. Moral of the story is, if you have access to the knowledge or the knowledge itself you can with some hunting in the right geography land some amazing gear for really cheap.
One suggestion I might offer is to see if a buddy can loan or give you a beater to play around with. Then you can better equip yourself to make informed buying decisions.