Here's the process in simple terms - feel free to ask more detailed questions about each of the steps.
The two things you'll need before setting up a website are a domain name, and web hosting:
Your domain name is how people find you on the internet - this is the thing that you type into your address bar. Generally, you'll rent this from a company for a fee - I'd recommend buying your domain name and hosting in the same place, at least to start with.
Your hosting is the parking space your website sits in. Every website needs a computer to live on, and nearly all sites are hosted on server-grade computers in a server farm. Once again, you'd rent this from a company and pay a fee per month for it.
So, your first order of business is to get these. Generally, you can buy them both in the same place - a lot of hosting packages include a domain name. Where you buy them from is the big question - there's literally thousands to choose from. For the sake of argument, let's use my host as an example, the UK-based Krystal Hosting, but I encourage you to find one that suits your needs and price bracket. What you'll need is a web host that offers PHP and MySQL - if they don't specify, they probably offer both.
To begin with, you'll want shared hosting - this means your website is stored on a computer with many others, but is much, much cheaper than the alternatives and perfectly good for a new forum. For playing around you can probably go with a base package. So, at this point you'll want to buy your domain name and hosting. After you buy them, they'll take between ten minutes and a few hours to set up and become accessible.
Afterwards, let's assume you haven't yet bought a XenForo license - so click Buy XenForo, and go through the steps of buying a basic XenForo license. Once you've done that, you'll be able to see the Your Licenses screen - click Download XenForo and make sure "Download the upgrade package only" is unticked, but agree to the license terms.
You'll now need to extract those files and open up the "upload" folder within.
Now, to get the files within onto your web hosting, you'll need an FTP program - either FileZilla for PC or Cyberduck for Mac will do. You'll need to open up your FTP program and connect to your server using the details provided by your host - they'll generally be in an e-mail you get when setting up the hosting, or alternately in your hosting control panel. You'll need a hostname (generally your new domain name, but not always), a username, and a password. If you can't find them, try using the same details you use to login to the control panel as the username and password.
Now it's case of finding where to upload the files: Generally, there will be a "public_html" or "www" folder in the hosting - if so, open that. If not, you're probably safe to upload the files in the first directory. Now it's a case of uploading the contents of the upload folder on your PC into that folder in the FTP client - just drag and drop.
After that process has finished, which might take a while, and your domain name is resolved, you can access the XenForo installation. Navigate to your domain name in your browser, which will take you to XenForo's installation wizard if all went well.
Next up, let's create the database. This is a slightly complicated procedure, because it varies by host, but you'll want to look in your control panel for the option to create a MySQL database. You'll need to specify a database name and a password at least - make the password long and very complicated, you'll need to make a note of it but you won't need it in day-to-day activities, so make it hard to crack. You'll then need to enter your database name, database password, and username into the respective fields in the XenForo installer - just follow the wizard through.
Lastly, after the wizard is finished, you're done installing! You can now play with your shiny new XenForo install - enjoy!