Generic shared hosting providers
About shared hosting
Shared hosting is an arrangement where your site is on a server with many other peoples’ sites, sharing a single pool of server resources -- storage space, memory, bandwidth, etc. Performance may not be as fast or reliable in a shared hosting environment as hosting your site on its own dedicated server because other sites on the server may experience a surge in traffic, or undertake some memory-intensive operations, that can take up a disproportionate amount of the shared resources. Hosting companies monitor sites in a shared hosting environment for that sort of behavior and intervene, requiring those sites to upgrade to a higher hosting tier that has more resources available if they want to continue in such a manner.
When a hosting service says that your site can have “unlimited” databases, bandwidth, storage, etc., that’s a little misleading: the hosting service lets you have what it deems to be a “reasonable” amount (that it won’t state upfront), but if you go over the unstated limits, they’ll contact you and ask you to reduce your usage or upgrade to a more expensive hosting plan. The cutoffs for those unstated limits may also change over time without any advance notification. This has happened to me while running a multisite installation on Bluehost -- for years it was fine, and then I got a notice saying I had too many database tables. After I removed a few test sites, I was good for a year or so, before I received another notice saying I had to reduce my database usage.
Recommended: Reclaim Hosting
One of the most widely recommended hosting services for digital humanities projects is Reclaim Hosting. Founded by people with higher-ed IT experience, Reclaim Hosting has more inexpensive pricing options than the typical shared hosting environment (starting at $25/year). They also offer considerably more (and much friendlier) technical support if you run into problems while you’re migrating or building your site, and they even offer free migration assistance.
Reclaim Hosting may not be a good fit for your project if you have a lot of multimedia (files, images, video, audio, etc.) that you want to store locally with your site, rather than embedding it from a 3rd party service like YouTube, etc. Their largest plan currently offers 100 GB of storage, and runs $100/year, a price comparable to other shared hosting services (but with the advantage of better tech support). It may also not be a good fit if you want to run a large multi-site installation to support multiple sites whose combined amount of storage would go over the 100 GB limit.
Reclaim Hosting uses cPanel as the interface for hosting accounts, and you can find information on installing Drupal using cPanel here. They also have Installatron for installing software packages including Drupal, which makes the process extremely easy. (They have step-by-step instructions for installing WordPress with Installatron, but installing Drupal works the same way.)
Other hosting options
There are countless shared hosting services available, and it’s not easy to wade through all the options and pick one. Drupal has a hosting page listing the hosting services that have sponsored the Drupal Association. This in and of itself doesn’t say anything about the quality of the hosting service, but it does attest to the fact that they are trying to get more business from people who use Drupal (which suggests that their infrastructure is set up in such a way that you can expect Drupal sites to perform reasonably well -- the same cannot be said for all shared hosting providers). It also suggests they have enough resources to sponsor the Drupal Association, and to be thinking about that kind of outreach, which is less likely for fly-by-night hosting companies that might not exist next year.
One thing to research with shared hosting services is whether the renewal price is the same as the sign-up price. Many shared hosting services offer very low prices (under $5/month) for the first year of service, but that price can double for subsequent years. Migration between services is a major hassle, leaving you paying the higher price. A higher subsequent-year price may be unavoidable, but you should at least find out what you should expect the next year before you sign up.
Most shared hosting companies use cPanel, and most have a “one-click installer” option for installing Drupal.