There's a crucial question that must be answered: are you able to completely manage your own linux server on your own? If you can, then you have plenty of options, and you just have to pick your price point.
But if you're like me, comfortable developing your wordpress blog, but you don't fully understand all that's required to configure, install, and secure a linux server, then nearly all of those options are useless. It's what keeps many people on low-end shared hosting (Bluehost), where they don't have to worry about anything on the server - they can just worry about their blog, and everything "under the hood" is taken care of.
If this is the case, then when you need more performance, you need to specifically look for "fully managed" hosting, whether on VPS or dedicated servers. And that's where your options get much slimmer (or much more expensive).
This issue of "server management" is the not-so-obvious lynchpin when you're shopping for hosting. Nearly all the services you'll find will just give you a linux server (virtual or literal), throw some basic control panel on it and say "good luck". If you don't know how to re-compile PHP with the proper extensions, you have to open a ticket, wait a long time, and pay through the nose for "advanced tech support", as most included support only covers failure of the actual server.
I've recently moved all of my personal and client hosting to Unixy, for one simple reason: they will do whatever I ask, whenever I ask it. They offer fully managed VPS and dedicated servers, and they really mean it. You can get a GUI control panel, but you don't even need one. You name it, they'll do it, no matter what it involves. I have yet to find any request that doesn't fall within their full-service guarantee. It's like having my own personal linux sysadmin on call 24/7, for no extra charge.
Typical interaction via email:
ME: Can I please get xdebug extension enabled?
UNIXY: (60 seconds later) This is in progress.
UNIXY: (5 minutes later) This is completed.
I'm completely serious. I've never waited longer than 5 minutes to get a response to any email, even at 3AM. And while this is obviously a simple request, I've asked them to write custom backup and cloning scripts for me, even help debug custom wordpress development. Literally anything you'd ask if you had your own full-time private linux sysadmin.
I have no affiliation with Unixy (and no paid affiliate link), I'm just a happy client. If you need more performance than shared hosting, but managing your own VPS is too scary, check them out.