That is awesome!
As for things to do, Trumpets under the Dome is pretty awesome.
This event list (which has everything going on during the weekend, so take a look) has it at 5:30pm on Saturday for Stanford.
As for other things, I would recommend checking out the Grotto and the Basilica which are both close to the Dome so its easy to do all three quickly. You will probably want to get a picture at Touchdown Jesus, but you could also try to get ones with First Down Moses and Fair Catch Corby.
When you get hungry you can stop at any of the student run tables selling food (where proceeds go to any number of different causes), but if you really want to experience a ND gameday you will go to the Knights of Columbus building where they sell steak sandwiches which have basically become a tradition on their own.
If the weather isn't great or you just want to sit down for a while you should head to the fan fest in the Joyce Center. They have food (and I think alcohol) plus games, activities, and TVs. If you do stop there, you should check out heritage hall which is just upstairs where they have all sorts of trophies and historical displays from all the different Irish sports teams including football. Depending on what time it is you may also be able to check out the basketball team scrimmaging on the main court which is in the same building.
As for crashing a tailgate, you will probably be able to find someone willing to let you crash if you are nice enough and dressed in Irish gear. I would recommend bring a case or two for yourselves just to show you aren't trying to mooch off others, but whoever you end up crashing with will probably force their drinks on to you anyway.
Final pieces of advice are to make sure you prep for the weather because it can be anything from really hot to really cold, rainy to dry, and different combinations of that throughout the day. Also, the stadium has instituted a new clear bag policy this year, so make sure your bag is allowed before trying to go into the stadium because it is a pretty strict guideline, so much so that even the bookstore shopping bags aren't allowed in.
I hope you have fun and let me know if you have any other questions!