Later That Day
The first thing to address is the transition area, where your bicycle and other gear have been sitting since morning. The rules differ, so be sure you know how your triathlon organizers manage the transition area.
Some triathlons do not permit anyone who is not still competing to enter the transition area until a certain time that day. Others require everyone to be out of the area by a certain time. All in all, it's best not to plan any activity on race day that will require you to get away immediately after the competition. There are also strict rules about leaving trash behind, so plan to bring a plastic sack for empty gel packs or energy/protein bar wrappers.
While you're relaxing and reflecting on your accomplishment, you will have time to return to the leaderboard. Most races update the standings continuously as finishers come in. Complete results are usually available on the Internet almost immediately.
Once you're done and have cooled off, stretched, started rehydrating, and put some food in your stomach, go back to the finish area and see if you know anyone, perhaps some of your training partners. Find someone to congratulate. Share hugs and war stories from your race.
If it's your first triathlon, there will never be another. Savor the experience. Take photos if you have a camera, and look around so you can remember the scene. There will be lots of thoughts running around in your head. Dismiss anything resembling recriminations — what you could have or should have done differently. There will be time for that later. For now, just relax and enjoy the feeling of achieving your goal.
If you are not under pressure to get away quickly, consider lending a hand to the organizers who put the race together. A significant number of them, probably the majority, will be volunteers, and they will definitely appreciate any assistance in tearing down the finish line and packing up the gear. Even if they don't need your help, you'll get credit for offering.
Once you're done and have stowed your bicycle and other paraphernalia, head for home or the hotel to clean up and plan for a brief nap. Sleep is a very good recovery tool. You don't have to stay in bed for a long time, but the rest will leave you refreshed and ready for more fun that evening.