It's a big decision! There are so many positives and negatives to consider when planning the technology platform that will run your website.
If price is the biggest factor, going with a hosted platform is often a great solution. These include platforms like Squarespace or Spacecraft, that offer an array of design starting points. The benefit is that it's often a low cost solution, as well as being fully stacked with the latest and current SEO optimizations, social media integrations, and nifty gallery features built in. The downside is that you don't actually own the site, you just own the content on it. You really just lease the software from the company. So you have no option on where it's hosted, nor can you pick it up and move it. If you stop your contract, your site is literally 'off.'
Custom building can happen in a variety of ways, and involve a variety of platforms. These range from the fast-cheap approach of customizing a Wordpress template, to the slightly more involved process of building a custom PHP interface. Other than Wordpress, which is the most popular platform right now, there is Drupal and Joomla who both offer many features. All of these are open source, so finding developers are plentiful. One downside is that it's really kinda hard to know exactly how good your developer is before you start (unless you can assess code), so sometimes you can get into a bit of a rabbit hole in hour after hour and round after round of edits.
Beyond that, sites need to be optimized for mobile performance. This doesn't require iOS or Android platform programming, but it does require planning. Some interactive projects need to actually be built on a native app platform.
The benefit of working with a design focused company is that you won't get locked in to any particular platform, and can be more confident your technology selection is right for your project. We believe it's best to have a solid strategy, plan, and even a pretty detailed set of wireframes prior to finalizing a technology decision if it's not predetermined by other factors.