You would go into whatever control panel your host provides, locate the MySQL section, then locate the link to open phpMyAdmin (hopefully, they have that). Who is your provider?
From there, you’d locate your database and export the tables. In most cases, this is done by clicking on the Export tab, then selecting all the database tables in the View Dump>Export column. I typically accept the default settings (except for selecting all the tables) and export using the Save as File option… I’m using version 2.9.2 of phpMyAdmin, so a newer version (or older) could vary a bit.
Save it to your desktop.
Then, use FTP to save all the custom files for your theme, your plugins and such. I keep a list of what files are different from the basic install as I tend to modify more than just the theme, but that’s me.
Then, create a new installation of WordPress – I use hosts that offer Fantastico, so this is an easy thing… You can opt to delete the old one and install a new one using the same folder, etc so nothing should change. Be sure to use the same admin name and password…
Once you’ve installed the new WP files, make sure that the site shows up (this won’t have your data in it, but make sure it works)
Now upload the custom template files and in your admin pages, select your theme.
Then, using phpMyAdmin again, get into that database. Select all the tables. There should be a Check All/Uncheck All link to do that. To the right of that link, there should be a menu that allows you to choose “drop”. Do it. You are only dropping empty tables, really.
Now you are ready to bring in your own data.
Using phpMyAdmin in your now empty database, click the Import tab. Browse out to your desktop and select the database your exported from the original WP installation. Click the Go button and keep your fingers crossed!
HTH. If you’ve not changed the folder where you’ve installed or the domain name, etc… this is all you should need to do. Check through the site to see if that fixed things.
The database tables contain everything about your site, from its name to url to post & page content to theme selection, etc