Upgrading is really easy, but it is different than installing.
Instructions are here, but I'll condense them for you:http://codex.wordpress.org/Upgrading_WordPress
1. Back up your database.
2. Use an FTP client or some other way to back up ALL the files in your old install onto your hard drive or where ever.
2.1 (Steps one and two are to protect you in case you screw anything else up. That way, you can always revert back to your old site.)
3. Disable ALL of your plugins through the wp dashboard. This is very important!
4. Upload the new WP files to your system. Keep in mind that any new files will overwrite the older files with the same name, so if you have made any changes to any wp core files, such as your rss feeds or whatever, you might want to *skip* uploading those. In other words, you only need to upload the files that have changed from 1.5 to 2+ and you should be able to find a list of those at the above link.
4.1 It is NOT neccessary to overwrite wp-config, anything in your wp-contents folder or anything in your languages folder if you are using a language other than english.
5. Once the new files are uploaded, make sure all your old themes and plugins are still there (in the wp-content folder) and if not, re-upload those from the backups you made in step 2
6. run upload.php. This is the step that differentiates installing from upgrading. There are a few new database tables that have been added to wp 2+, and running upgrade.php will make sure those get added.
7. Go to your admin panel and active your plugins again. Your site will be shiny and new but it should still look the same.
Most problems will be caused by either forgetting to upload a file or by plugins that aren't compatible with the newest version. These are the first things to check if you have an issue with upgrading.
Post back if you encounter any problems.