Mazel tov! You must be getting close to your conversion if you're thinking about how to celebrate it!
My experience so far is that orthodox converts don't celebrate nearly as much as liberal converts. Make your interpretations as you will.
Nearly everyone has (or tries to have) a special meal with some friends and family. You will hopefully receive cards and congratulations from your friends and members of your community. Some may even make a contribution to the synagogue or to a charity on your behalf. Remember to write "thank you" cards or otherwise thank these kind people!
For men, things seem to be pretty standard: you'll almost certainly get an aliyah on your first Shabbat as an "official Jew." Some people choose to treat it almost like a bar mitzvah, and they'll spend the time training to lein (leyn, however you want to spell it) the Torah and lead at least one service on Shabbat. Anything beyond that is up to you!
For women, things are anything but standard. Be creative and do what makes you happy! Personally, I'd like to host something for myself akin to a Shabbat Kallah, where people gather on Shabbat afternoon and celebrate with food, games, and brachas (brachot, blessings)!
What I don't suggest? Don't immediately start calling the shadchans (matchmakers) to start looking for your beshert (soulmate)! There's a great Jewish idea: don't mix simchas. Enjoy this simcha for a little bit, at least a few weeks but preferably a couple of months, before starting the pain that can be the shidduch process!
Enjoy the glow of being newly-converted! Share brachot and kind words freely. Help others feel the joy you (hopefully) feel! And if your conversion was anticlimactic, let yourself "catch" the enthusiasm from those closest to you! Science shows that if you smile when you're not happy, you'll trick your brain into thinking you really are happy! You're like a kallah (bride) regardless of whether you're male or female, and you should share your particularly fresh closeness with Hashem by making a bracha for (or otherwise davening for) the people around you!