Thankfully, Delicious provides a very easy export method for getting all your bookmarks into a browser-friendly HTML file. Just head into Delicious' Settings and hit Export / Backup Bookmarks. You can choose to include your tags and notes with the bookmarks as well, but these won't import into your browser; you'll only be able to view them by opening up the file that Delicious gives you. Hit Export and move over to your browser. Note that depending on your browser, you may need to change the resulting file's extension from .htm to .html first.
Open up your browser's bookmark manager and look for an import option. In Google Chrome, you'll find it under Organize > Import Bookmarks; in Firefox it's under the button with the star icon. When prompted, navigate to the HTML file that you downloaded from Delicious and your browser'll put those bookmarks into its bookmark manager.
Apart from the lack of tags and notes from Delicious — which is to be expected — they should all be imported just fine. If you had put a lot of effort into organizing your Delicious bookmarks, there's no way around it — you'll have to separate them all into folders manually again. But at least you haven't lost those links, right?