Start A New Game
Once you reach level 20, one of the most obvious things you can do is begin a new game with a new character. This is also the most basic thing you can do without having to modify your game or character. To make things more interesting, create a character with a class and a set of abilities that you've never tried before. Be a mage instead of a warrior. Use brute strength as a fighter instead of sneaking around as a thief.
Become A Vampire
An interesting aspect of "Oblivion" is that you can be bitten by, and become, a vampire. When you are turned into a vampire, many things about your character change: your abilities, your weaknesses and even your appearance. Becoming a vampire at a very low level can make "Oblivion" an entirely different game. You become sensitive to sunlight and must drink blood to survive. This change also makes it difficult to talk to other characters. Being a vampire makes the game far more challenging, as you will have more things to worry about than your character's general health.
Complete All Side Quests
"Oblivion" has many different side quests and guild quests available to those who finish the standard storyline. Locating them all can be just as challenging as completing them successfully. Completing all the side quests can also draw you into secret and hidden locales that you may not have been able to find otherwise. Most of the side quests also have ongoing stories and intertwining plots that help you better understand the main storyline.
Modify Your Game
"The Elder Scrolls" series of games boasts one of the largest modding communities of fantasy role-playing game titles. Modifying your game directly through programmed and custom content can allow you to add different graphics, new quests and stories and whole new areas to explore, and can even let you select a level of difficulty that far surpasses hard mode. "The Elder Scrolls Nexus" carries many types of "Oblivion" mods that are available for download.