Choose "File System" and have it export your website to some new directory in your filesystem.
Then inspect it to ensure everything's in order and present. Open up your web.config file and change things like connection strings and system-specific settings to match your host's, then upload it using an FTP client to the right directory within your webspace.
The FTP client built-in to Windows Explorer shoud be fine.
Some hosts don't let you define custom application scopes, in which case you'll want to move the \bin directory and web.config file (assuming you're using "ASP.NET Web Applications" and not "ASP.NET Websites") to the root of your website's public folder
(often called "public_html" or "htdocs") if your application won't exist in the root of your website.
As for your database, you'd use SSMS to connect to your host's SQL server and upload the bootstrap data into your database (in case they don't allow you to run a RESTORE you'll have to use DTS to restore the structure and do a batch import of data).
As for the stuff you've posted:
The FTP option would work, but you don't get any second-chances to check what you're uploading. I'm not a fan of the Publish feature introduced in VS2005 which was added mainly for the benefit of beginners uses using VWD who wouldn't be familar with the
"Web Applications" way of doing things. Note that the FTP client built-in to VS2008 doesn't support FTPS (FTP over SSL) or SFTP (SSH FTP).
The "Remote Site" is just a byword for FPSE/WebDAV uploads. FPSE is de-facto deprecated now (the last version of FPSE was in 2002 and it doesn't offer any compelling reasons to install beyond making life simpler in the short-term for users of FrontPage).
I understand that the "Remote Site" upload feature might work using plain-old WebDAV without FPSE, but very few hosts offer WebDAV as a means of managing website files. WebDAV is more secure than FTP because it doesn't send plaintext passwords.
The Remote Site location would be a WebDAV address, which is a normal URL (e.g. http://www.mywebsite.com/webdav) except the server is configured to recognise the URL as being a webDAV-enabled location and so enables file management features. WebDAV is a
powerful and underused/underrated feature, it's also the underpinning of SVN, which is testament to how good it is.