It's a good question you're asking. Finding the right host is one of those things that is different for everyone because chances are, your WP site is really different from everyone else's.
So it sounds like you've grown a site on shared hosting, and the time has come to upgrade to a more scalable host that can grow with you, but not add any hassle to your life when your traffic spikes. For the record, I still have a bluehost account where I host a few sites for my family. I've always loved their service. For you, it sounds like managed hosting is the way to go. It doesn't sound like you're interested in running your own VPS ;-)
Now, I'm horribly biased since I'm the brand ambassador for WP Engine, so please verify what I say here for yourself. My goal is that you find the host that makes the most sense for your site, AND that you're actually *happy* about your hosting provider.
<Now prepare yourself to hear all about WP Engine (Engage Marketing Speak!)>
WP Engine provides very scalable hosting. We regularly host traffic spikes of 10x normal, and then some, without any downtime or loss of performance. We only host WordPress, so we have tailored our stack to make this possible. We will never throttle your site if you exceed your traffic for the month. If you have a traffic spike, that usually means you're being successful, and we want to make sure your website is prepared for that success! Of course, if you need to upgrade to handle continually growing traffic, that's a really simply process, and we'll take care of all the complicated stuff for you.
The thing you'll like most is our tech support. We only have WordPress experts on staff to answer your questions. And while we can't provide custom development for your theme as part of support, we know WordPress, and can solve problems on far more than just the server side. You'll never be stranded without help on your site.
Again, don't take my word for it. Here's a list of other managed hosts you should look at: ZippyKid, WebSynthesis, and Pagely. I know the people from these companies, and we all spend time together at conferences. They're all amazing folks who are providing hosting to thousands of WordPress sites.
So you don't have to believe me, a quick google will show you customer reviews like these:
Ok, I'm done. /marketing speak.
Please let me know if you have more questions. I'm happy to answer them and help you get settled at the right host, whether that's with us or elsewhere.