I have experience with a few HTC's and 4-5 Samsung Galaxy models.
Galaxy S3 still runs great (my daughter has it now). Wife has a Galaxy S5 and if you can cope with the close-to-brick size of it, you'll love it. She also has a Galaxy S3 Mini as her business phone, which I honestly find more handy. But it's also significantly inferior in terms of tech specs.
Me, I went from the Galaxy S3 to the Galaxy S4 Mini. About half the price of the S5, but significantly smaller. I have a problem with brick-size phones.
Anyway, it handles all games I've tried flawlessly. 8GB default storage, which I increased to the maximum of 8GB + 64GB memory card. It's quite fast and chances are that getting anything newer is pretty much overkill. Overall I love it. I have weeks of music on it for when I hit the gym and I can even download the occasional movie on it, to watch when stuck waiting outside the store, while the wife is shopping inside.
Anyway, you can get it for less than $200, depending on where you shop. The only alternative to the S4 Mini I'd consider is the S5 Mini. However the differences are pretty slight, so IMO paying almost double for the S5 Mini is not worth it. The only property it has, that I wish my S4 Mini had too, is that it's waterproof. So if you tend to carry your phone in your breast pocket and bend over when you flush the toilet, then the S5 Mini is for you.
BTW, search Google for some reviews for these. Plenty of reputable sites test them and post detailed reviews.
Hope this helped.
